
    wg,                        d dl mZmZmZ d dlmZmZ d dlmZ d dl	m
Z
 d dlmZmZ d dlmZmZmZ d dlmZ d dlmZ d d	lmZmZ d d
lmZ d dlmZmZmZmZ d dl m!Z!m"Z"m#Z# d dl$m%Z%m&Z&  ed      Z' edd      \  Z(Z) ed      \  Z*Z+Z,d Z-d Z.d Z/e#d        Z0d Z1d Z2d Z3d Z4d Z5d Z6d Z7d Z8e"d        Z9d  Z:d! Z;d" Z<d# Z=y$)%    )FunctionLambdaexpand)IRational)Eq)S)Symbolsymbols)rfbinomial	factorial)Abs)sqrt)cossin)factor)rsolversolve_hyperrsolve_polyrsolve_ratio)raisesslowXFAIL)abyzn,kT)integerzC0,C1,C2c                     t        g ddt              dk(  sJ t        g ddt              dk(  sJ t        dt        dz   gt        t              dk(  sJ t        ddgt        t              t        t        dz  t        z
  dz  z   k(  sJ t        t         dz
  t        gdt              t        t        z  dz
  k(  sJ t        dt        z  dz
  dgdt        z  dz   t              dk(  sJ t        ddgt        dz  t        d	z  z   t              t        t        d	z  dz  z
  t        dz  dz  z
  t        dz  d
z  z   t        d
z  d
z  z   dt        dz  z  d	z  z   k(  sJ y )Nr!      r   r"   r!                  )r   nC0     d/home/mcse/projects/flask/flask-venv/lib/python3.12/site-packages/sympy/solvers/tests/test_recurr.pytest_rsolve_polyr.      sD   {Aq)Q...{Aq)R///AE{Aq)Q...Aw1%q!taxl)::::Q{Aq)RTAX5551q!}acAgq1R777Aw1q!tQ/
QTAX1q 1a4!8+adQh61a4!CD D Dr,   c            	         t        dt        dz  z  t        dz  z   dt        z  z   dz
  dt        dz  z  t        dz  z   dt        z  z
  dt        dz  z  dt        dz  z  z
  dt        z  z
  dz
  dt        dz  z  d	t        dz  z  z   d
t        z  z   dz   gdt              } | t        dt        z  dz
  z  t        dz  dz
  z  dz  k(  sJ y )Nr'   r#   r"   r(         	            r   )r   r)   r*   )solutions    r-   test_rsolve_ratior8   !   s    R1Wq!t^ac1A5qAv1}qs7J
1a4"QT'BqD 1$a1fr!Q$w&6A&=&ACDEqJHr1Q37|QTAX.q0000r,   c            	      
   t        g ddt              t        t        j                  t        j                  t        d      z  z
  t        z  z  t        t        j                  t        j                  t        d      z  z   t        z  z  z   t        t        j                  t        j                  t        d      z  z
  t        z  z  t        t        j                  t        j                  t        d      z  z   t        z  z  z   fv sJ t        t        dz  dz
  dt        z  dz
  dgdt              t        t        t        d      t              z  t        t        t        d       t              z  z   t        t        t        d      t              z  t        t        t        d       t              z  z   fv sJ t        t        dz  t        z
  dt        z  dz
  dgdt              t        t        t        t              t              z  t        t        t        t               t              z  z   t        t        t        t              t              z  t        t        t        t               t              z  z   fv sJ t        dt        z  t        dz   z  t        dz   dt        z  z
  dz   t        dz
  gdt              t        t        t              z  t        dt        z  z  z   k(  sJ t        t        dz   dt        z  dz    dt        dz  z  d	t        z  z   d
z   z  t        dz   gdt              dk(  sJ t        t         dz
  ddgdt              dk(  sJ t        ddgt        t              j                         t        t        dz  dz  z   t        dz  z
  k(  sJ t        ddgdt        z   t              j                         t        t        dz  dz  z   t        dz  z   k(  sJ t        ddgdt        t        dz  z   z  t              j                         t        t        dz  z   t        z
  k(  sJ t        t         dgdt              j                         t        t        t        z  z  k(  sJ t        t         ddgdt              j                         dt        z  t        z  t        t        dz  z  z  t        t        t        dz  z  z  z   k(  sJ t        g ddt              j                         t        t        dd      t        d      t        z  dz  z
  t        z  z  t        t        dd      t        d      t        z  dz  z   t        z  z  z   k(  sJ t        ddt        z  t        z  dt        z  z
  dgdt              dk(  sJ y )Nr    r   r&   r#   r0   r"   r'      3   '   r!   )r"   r"   r"   )r   r)   r*   r	   Halfr   C1r   kr   r   r   r   r   r+   r,   r-   test_rsolve_hyperr@   '   sx   Q*
AFFQVVDG^#a''"affqvvd1g~.E-I*II
AFFQVVDG^#a''"affqvvd1g~.E-I*II/   
 A2a4!8Q/A6
2d1gq>Br47(A..
2d1gq>Br47(A..;   
 A2a4!8Q/A6
2d1gq>Br47(A..
2d1gq>Br47(A..;   
 	
1a!eq!teackAoq1u-q!58:9Q<"QPQT'8QR R R 	
Q!A#'
Bq!tGbdNR/0!a%8!Q@CDE E E !aQA.!333QA&--/21Q;13DDDDQQ*113rAqDF{QqS7HHHHQA1Hq188:b1a4i!mKKK!Q!$++-AqD888!Q
Aq)002rAgbjQqS6IBqSTUVSVxK6WWWW	1a(//1
HROd1gaik)A--HROd1gaiPQk4QTU3U0UUV V V BqDFQqSL!,a3q888r,   c                  N    t        t         t        dz   gt        t              J y )Nr#   )r   r)   r+   r,   r-   test_rsolve_ratio_missedrB   O   s#     !QUQ*666r,   c                 T     t         fdt        t                     D              S )z9Compute RHS of recurrence in f(n) with coefficients in c.c              3   d   K   | ]'  }|   j                  t        t        |z         z   ) y w)N)subsr)   ).0icfs     r-   	<genexpr>z"recurrence_term.<locals>.<genexpr>X   s)     <qtAFF1a!e$$<s   -0)sumrangelen)rH   rI   s   ``r-   recurrence_termrN   V   s    <eCFm<<<r,   c            
         t         t         dz   t         dz  t         dz  t         dz  t         t         dz  z   dt         z  dt         dz  z  z   dt         dz  z  z
  dt         dz  z  z   dt         dz  z  z
  g} d	dgg d
g dt          dgt         dz  t         z
  dz   dgg}| D ]  }|D ]  }t        ||      }|j                  t               rt        ||t               |k(  sJ |j	                  t               sPt        |      dk  s_t        ||t               j                  t        t        d      g d            j                         |k(  rJ   y)zSome bulk-generated tests.r"   r#   r'   r%      4      r&   r0   )r0   r!   r"   )r!   r"   r"   r!   r"   zC:3)r   r   r   N)r)   rN   is_polynomialr   is_hypergeometricrM   r   rE   zipr   r   )funcscoeffsprH   qs        r-   test_rsolve_bulkrZ   [   sP   Q1adAqD!ad(BqD2ad7NQ	1F 5!Q$w5AqD5! #EAw%6!Q!Q$	C
CB F ` 	`A1%Aq!"1a+q000""1%#a&A+#Aq!,11#geni2PQXXZ^____	``r,   c                      t        t        dz  t        z
  dz   dgt        t        dz  t        z
  dz   z  t        z   dz   t              t        k(  sJ y )Nr#   rR   r"   )r   r)   r+   r,   r-   test_rsolve_0_sol_homogeneousr\   l   sG     AB*Aq!tax"},=,AA,EqIQNNNr,   c                     t        t        dz         t        t        dz         z
  t        t              z
  } t        d      t        j                  t        j                  t        d      z  z   t        z  z  t        d      t        j                  t        j                  t        d      z  z
  t        z  z  z
  }t        | t        t                    t        t        j                  t        j                  t        d      z  z
  t        z  z  t        t        j                  t        j                  t        d      z  z   t        z  z  z   t        t        j                  t        j                  t        d      z  z
  t        z  z  t        t        j                  t        j                  t        d      z  z   t        z  z  z   fv sJ t        | t        t              ddg      |k(  sJ t        | t        t              ddd      |k(  sJ t        | t        t              t        d      dt        d      di      |k(  sJ t        t        t              t        t        dz
        z
  t        t        dz
        z
  t        t              ddg      |k(  sJ t        t        t        t              t        t        dz
        t        t        dz
        z         t        t              ddg      |k(  sJ | j                  t         t        t        t        | t        t                    j                  t        t                          j                         dk(  sJ t        dz
  t        t        dz         z  t        dz  dt        z  z   dz
  t        t        dz         z  z
  dt        z  t        dz   z  t        t              z  z   } t        t        t              z  t        dt        z  z  z   }dt        t              z  ddt        z  z  z   }t        | t        t                    |k(  sJ t        | t        t              g       |k(  sJ t        | t        t              i       |k(  sJ t        | t        t              ddg      |k(  sJ t        | t        t              ddd      |k(  sJ t        | t        t              t        d      dt        d      di      |k(  sJ | j                  t         t        t        t        | t        t                    j                  t        t                          j                         dk(  sJ t        t              t        t        dz
        z
  dz
  } t        | t        t              t        d      di      dt        z  k(  sJ t        | t        t              t        d      di      dt        z  dz   k(  sJ t        | t        t              t        d      dt        d      di      J | j                  t         t        t        t        | t        t                    j                  t        t                          j                         dk(  sJ dt        t        dz
        z  t        t              z
  dz
  } t        | t        t              t        d      di      dt        z   dz  t        j                  z   k(  sJ t        | t        t              t        d      di      dt        z  dz  t        j                  z   k(  sJ t        | t        t              t        d      di      ddt        z  z  dz  t        j                  z   k(  sJ | j                  t         t        t        t        | t        t                    j                  t        t                          j                         dk(  sJ t        t              dt        z  t        t        dz
        z  z
  } t        | t        t                    t        t        t              z  k(  sJ | j                  t         t        t        t        | t        t                    j                  t        t                          j                         dk(  sJ t        t              dt        z  t        t        dz
        z  z
  dz
  } t        | t        t                    J dt        t        dz
        z  dt        z
  t        t              z  t        z  z   } t        | t        t              t        d      di      dt        dz
  z  t        z  k(  sJ t        | t        t              t        d      di      dt        dz
  z  t        z  dz  k(  sJ t        | t        t              t        d      di      dt        dz
  z  t        z  dz  k(  sJ | j                  t         t        t        t        | t        t                    j                  t        t                          j                         dk(  sJ t        dz
  t        dz
  z  t        t        dz         z  t        dz   t        dz   z  t        t              z  z
  } t        | t        t              t        d      dt        d	      d
i      t        t        dz
  z  t        dz
  z  k(  sJ t        | t        t              t        d      dt        d	      di      t         t        dz
  z  t        dz
  z  dt        z  z  k(  sJ | j                  t         t        t        t        | t        t                    j                  t        t                          j                         dk(  sJ t        t        t        t        dz         t        t        t              z        t        t              t        d      t        i      j                         t        t        z  k(  sJ t        t        t              t        t        t        dz
        z  z
  t        t              t        d      t        t              t        t        z   z  t        d      t        t        t        z
  z  i      j                         t        t        dz  dz   z  t        t        t               t        z  z  z
  k(  sJ dt        dz  z  dt        z  z   dz
  t        t        dz
        z  d	t        dz  z  dt        z  z
  dz   t        t              z  z   } t        | t        t              t        d      t!        dt        z  dz   d      i      }ddt        z  z  t        z  dt        z  dz
  dz  z  dt        z  dz   z  dz  }t#        t%        |d            |k(  sJ t        t        t              t        t        t        dz         t        t        dz
        z   z  dz  z   t        t                    }t'        |      dk(  sJ t        t        dz   t        t              z  t        t                    J t        t        dz   t        t              z  t        dz   t        t        dz         z  z   t        dz   t        t        dz         z  z   t        t                    	 J t        t        t              t        t        dz         z   dt        z  z   dt        z  z   t        t                    dt        z  t        z  dt        z  dz  z
  dt        z  d	z  z
  k(  sJ y )Nr#   r"   r&   r   )r   r"   r'   r(   r%      ir!   i    rR   r3   T)funcz@C0*((-sqrt(1 - a**2) - 1)/a)**n + C1*((sqrt(1 - a**2) - 1)/a)**n)r   r)   r   r	   r=   r   r*   r>   r   rE   r   r?   simplifyr   r   r   r   r   r   str)rI   hgynsols        r-   test_rsolverh   r   s   	!a%1QU8ad"AQ!&&a.(1,,
q'166AFF47N*Q.
.	/A !QqT?
AFFQVVDG^#a''"affqvvd1g~.E-I*II
AFFQVVDG^#a''"affqvvd1g~.E-I*II   
 !QqTAq6"a'''!QqTqQ<(A---!QqTAaD!QqT1-.!333!A$1q5/Aa!eH,adQF;q@@@"QqT1QU8aAh./11v>!CCC66!VAva133Aq9:;DDF!KKK	
Q!a%AqD1Q3JNAa!eH44qsAE{1Q47GGA
9Q<"QT'!A
9Q<!AqD& A!QqT?a!QqT2!###!QqT2!###!QqTAq6"a'''!QqTqQ<(A---!QqTAaD!QqT1-.!33366!VAva133Aq9:;DDF!KKK	!qQx!A!QqTAaD!9%1,,,!QqTAaD!9%1q000!QqTAaD!QqT1-.66666!VAva133Aq9:;DDF!KKK	!AE(
QqTAA!QqTAaD!9%!Q$q166)9999!QqTAaD!9%Aa!&&888!QqTAaD!9%1a4AFF)::::66!VAva133Aq9:;DDF!KKK	!qs1QU8|A!QqT?b1o---66!VAva133Aq9:;DDF!KKK	!qs1QU8|aA!QqT?"""	!AE(
a!eQqT\!^#A!QqTAaD!9%QUA555!QqTAaD!9%QUAa777!QqTAaD!9%QUAa77766!VAva133Aq9:;DDF!KKK	
QQ!a% AEAE?1Q4#77A!QqTAaD!QqT2./1a!e9a!e3DDDD	1Q4!A$1Q4%'+,"a!e*a!e*<bAY*FG G G 66!VAva133Aq9:;DDF!KKK"Qq1uXq1v&!qtQi8AACq!tKKK!A$1QqS6/!A$qT47AE?AaD!QU)466>hj!aL1tAwh]?*+ + + 
QTBqD2	qQx'1QT6BqD=1+<ad*BBA	1Q4!A$1q! 45	6B
ac(1*acAg\
!1Q37
+B
.C&$'(C///
11QU8aAh./111Q4
8Cs8YYYY1q5!A$,!%---AE1Q4<1q5!AE("22a!eQq1uX5EEqtL   !A$1q5/AqD(1a4/162'"*q!tAv:MPQSTPTUVPV:VVVVr,   c                       t        d       t        t        d        t        t         fd       t        t         fd       t        t        d        t        t         fd       t        t        d        y )Nxc                  t    t        t        t              t        t        dz         z
  t        t                    S Nr"   )r   r   r)   r?   r+   r,   r-   <lambda>z$test_rsolve_raises.<locals>.<lambda>   s!    vadQq1uXoqt< r,   c                  p    t        t        t              t        t        dz         z
    t                    S rl   r   r   r)   rj   s   r-   rm   z$test_rsolve_raises.<locals>.<lambda>   s"    vadQq1uXoqt< r,   c                  p    t        t        t                t        dz         z
  t        t                    S rl   ro   rp   s   r-   rm   z$test_rsolve_raises.<locals>.<lambda>   s"    vadQq1uXoqt< r,   c                      t        t        t              t        t              t        t        dz         z  z
  t        t                    S rl   )r   r   r)   r   r+   r,   r-   rm   z$test_rsolve_raises.<locals>.<lambda>   s+    vadT!WQq1uX-=&=qtD r,   c                      t        t        t              t        t        dz         z
  t        t                d      di      S )Nr"   r   ro   rp   s   r-   rm   z$test_rsolve_raises.<locals>.<lambda>   s,    vadQq1uXoqtadAYG r,   c                      t        t        t              t        t        dz         z   dt        z  z   t        t              z   t        t                    S )Nr"   r#   )r   r   r)   r   r+   r,   r-   rm   z$test_rsolve_raises.<locals>.<lambda>   s3    vadQq1uXo1&<s1v&EqtL r,   )r   r   
ValueErrorrp   s   @r-   test_rsolve_raisesrv      sN    A
:<=
:<=
:<=
:DE
:GH
:LMr,   c            	         t        t        dz         t        t        dz         z
  t        t              dz  z   } t        | t        t                    dt         dz   z  t        z  t        z  dt         z  t        z  z   k(  sJ t        | t        t              t        d      dt        d      di      ddt        z
  z  t        z  k(  sJ y )Nr#   r"   r%   r   )r   r)   r   r>   r*   rI   s    r-   test_issue_6844ry      s    	!a%1QU8ad1f$A!QqT?a1"q&k"nQ.aR;;;;!QqTAaD!QqT1-.!a!e*Q,>>>r,   c                     t        dd      } t        dd      }t        t              d| z  t        |      z  t        t        dz
        z  z
  | dz  t        t        dz
        z  z   }t	        |t        t                    t
        | t        |      t        t        t        |            z  z
  z  t        z  z  t        | t        |      t        t        t        |            z  z   z  t        z  z  z   k(  sJ y )NrT)positivetheta)realr#   r"   )
r
   r   r)   r   r   r*   r   r   r   r>   )r{   r}   rI   s      r-   test_issue_18751r      s    sT"A7&E	!q1us5z!Aa!eH,,q!taAh>A!QqT?
As5zAc#e*o--.22RCJ3sSXz?IZ<Z9[^_8_5__` ` `r,   c                     t        t        t        dz         t        t        dz         z
  t        t        dz         z
  t        t              z   t        t                    dt        z  t        z  t        z   t
        t        z  z   k(  sJ t        t        t        dz         dt        t        dz         z  z   dt        t        dz         z  z   t        t              z   t        t                    j                         dt        z  t        z  dt        z  t        z  t        z  z
  dt        z  t
        z  t        dz  z  z
  k(  sJ t        t        t              dt        t        dz
        z  z
  dt        t        dz
        z  z   dt        t        dz
        z  z
  t        t              g d      ddt        z  z  t        z
  dz
  k(  sJ t        t        t        dz         dt        t        dz         z  z
  dt        t              z  z   t        t              t        d      dt        d      d	t        d      d
i      d
t        z  dt        z  z   k(  sJ y )Nr'   r#   r"   r!   r&   r%   )r"   r'   r6   r   r6   r0   )r   r   r)   r>   r*   C2r   r+   r,   r-   test_constant_namingr      s   !AaC&1QqS6/AacF*QqT11Q48R!GBJObQRd<RRRR!AaC&1QqS6/!AacF(*1Q4/16==?B72:QSVWPWXZPZ[\P\C\`bef_fgi_ijkmnjn_nCnnnn!A$1QU8#a!a%j01Qq1uX:=ad7KqQRTUQUvXYz\]~]]] !AaC&1QqsV8#a!f,adQqT!QqT!QqT"4MNSUXYRY\]^_\_R____r,   c                     t        t              dt        t        dz         z  z   dt        t        dz         z  z
  dt        t        dz         z  z
  dt        t        dz         z  z   dt        t        d	z         z  z   dt        t        d
z         z  z
  dt        t        dz         z  z
  t        t        dz         z   } t        | t        t                    J y )N   r"   i  r#   iD  r'   i  r%   r&   r(      r6   )r   r)   r   rx   s    r-   test_issue_15751r      s    	!r!AE({S1q5\)D1q5M9D1q5MIDQRSTWXSXQYMY\_`abcfgbg`h\hhkmnopqtupunvkvvyz{|  @A  |A  zB  	BA!QqT?&&&r,   c            
         dt        t              z  dt        t        dz         z  z   dt        t        dz         z  z   dt        t        dz         z  z   } t        | t        t                    }t        dd	t	        d      dz  z  z  d
z  ddt        d      t        z  z   ddt        d      z  z   t	        d      dz  z  z  z   dt        d      t        z  z
  z  ddt        d      z  z   t	        d      dz  z  z  dz  z   dt        d      t        z  z
  ddt        d      z  z   t	        d      dz  z  z  z  t        z  z  t        dd	t	        d      dz  z  z  d
z  ddt        d      t        z  z
  ddt        d      z  z   t	        d      dz  z  z  z   dt        d      t        z  z   z  ddt        d      z  z   t	        d      dz  z  z  dz  z   dt        d      t        z  z   ddt        d      z  z   t	        d      dz  z  z  z  t        z  z  z   t        dd	t	        d      dz  z  z  d
ddt        d      z  z   t	        d      dz  z  z  z  t	        d      dz  z
  ddt        d      z  z   t	        d      dz  z  dz  z   t        z  z  z   }||k(  sJ |j                  t        dt        dt        dt        di      j                         }t        |dz         dk  sJ y )Nir%   r"   r(   r#   .   r'   V   r2   E   ir!   iln i  i#  i^     i  i      gYB?gvIh%<=)r   r)   r   r*   r	   r   r   r>   r   rE   evalfabs)rI   rg   expectedes       r-   test_issue_17990r      s   AaD1Qq1uX:!AE(
*R!a%[8A
AaD/CBrAaDF|OB&#d1gai&T$ZCQ46B# 1# +#%&a]*45:Sd^5K	
1a5** DGAIDJ(>!A$q&'IIK   BrAaDF|OB&#d1gai&4	dL CtAvB 1 +!"T!WQY*016T$Z1G	
1a1** DGAIDJ(>!A$q&'IIK   3rAaDF|#RT$Z)?1Q46(J%JK	!RDdO+qtAv6s:;=>? ??H (??"aQAq!,-335Aq$$%---r,   c                  $   t        d      }  | t        dz          | t        dz         z
   | t        dz         z
   | t              z   }t        | | t                    dt        z  t        z  t        z   t
        t        z  z   k(  sJ  | t        dz         d | t        dz         z  z   d | t        dz         z  z    | t              z   }t        | | t                    dt        z  t        z  dt        dz   z  t        z  t        z  z   dt        dz   z  t
        z  t        dz  z  z   k(  sJ t         | t              d | t        dz
        z  z
  d | t        dz
        z  z   d | t        dz
        z  z
   | t               | d      d | d      d | d      d	i      ddt        z  z  t        z
  dz
  k(  sJ t         | t              d | t        dz
        z  z
  t        z
   | t               | d      di      ddt        z  z  t        z
  dz
  k(  sJ y )
Nr   r'   r#   r"   r!   r&   r%   r   r6   )r   r)   r   r>   r*   r   )r   eqeq2s      r-   test_issue_8697r     s   A	
1q5Aa!eH	qQx	'!A$	.B"adQwrzBA5555
AE(QqQxZ
!Aa!eH*
,qt
3C3!!GBJ"Ar)!++rQUmB.>q!t.CCD E D !A$1QU8#a!a%j01Qq1uX:=A$1q!A$1Q43589!Q$
QG G G !A$1QU8#a'1!ay9Qq!tVaZ!^KKKr,   c            	         t        t              t        t        dz
        z
  dt        t        dz
        z  z
  dt        z  z
  } t        | t        t                    dt        z  t        z  dt        z  t        z  z   t        z
  t        dd      z
  k(  sJ t        | t        t              t        d      dt        d      di      dt        z   dz  ddt        z  z  z   t        z
  t        dd      z
  k(  sJ t        dt        t              z  t        t        dz         z   t        z   dz
  t        t                    dt        z  t        z  t        z   k(  sJ y )Nr"   r#   r!   r&   r   r0   )r   r)   r   r*   r>   r   rx   s    r-   test_diofantissue_294r     s   	!qQx!Aa!eH*$qs*A!QqT?rAgbj1a472Q6!QGGGG!QqTAaD"adA./b1WHQJ1a44G45568@A5G H H H "QqT'Aa!eH$q(1,ad3q!tBw{BBBr,   c            
      4   t        d      } t        t         | t              d | t        dz
        z  t        z          | t                    dt        z  t        z  t        z
  dz
  k(  sJ t        t         | t        dz         d | t              z  t        dz  z   dz          | t                    dt        z  t        z  t        dz  z
  dt        z  z
  dz
  k(  sJ t        t         | t        dz         d | t              z  t        dz  z   dz          | t               | d      di      ddt        z  z  dz  t        dz  z
  dt        z  z
  dz
  k(  sJ t        t         | t              d | t        dz
        z  dt        dz  z  z          | t                    dt        z  t        z  dt        dz  z  z
  dt        z  z
  d	z
  k(  sJ t        t         | t              d | t        dz
        z  t        dz  z          | t                    dt        z  t        z  t        dz  z
  dt        z  z
  d
z
  k(  sJ t        t         | t              d | t        dz
        z  t        z          | t               | d      di      ddt        z  z  t        z
  dz
  k(  sJ y )NrI   r#   r"   r%   r   r   r'   rR   r2   r(   )r   r   r   r)   r*   rx   s    r-   test_issue_15553r      s"   A"QqT1Qq1uX:>*AaD1QT"Wq[1_DDD"Qq1uXq1v1}q011Q48AqDGadNQqS<PST<TTTT"Qq1uXq1v1}q011Q4!A$CqAvaxRSUVRVYZ[\Y\G\_`G````"QqT1Qq1uX:!Q$./16!Q$r'AadF:JRPQT:QTV:VVVV"QqT1Qq1uX:1,-qt41R!Q$18Lq8PPPP"QqT1Qq1uX:>*AaD1Q4)<!Q$
QNNNr,   N)>sympy.core.functionr   r   r   sympy.core.numbersr   r   sympy.core.relationalr   sympy.core.singletonr	   sympy.core.symbolr
   r   (sympy.functions.combinatorial.factorialsr   r   r   $sympy.functions.elementary.complexesr   (sympy.functions.elementary.miscellaneousr   (sympy.functions.elementary.trigonometricr   r   sympy.polys.polytoolsr   sympy.solvers.recurrr   r   r   r   sympy.testing.pytestr   r   r   	sympy.abcr   r   r   r)   r?   r*   r>   r   r.   r8   r@   rB   rN   rZ   r\   rh   rv   ry   r   r   r   r   r   r   r   r+   r,   r-   <module>r      s    : : , $ " / N N 4 9 ? ( P P 4 4 SMud#1Z 
B
D1%9P 7 7=
`"OZWzN?`` ' '
. LCOr,   