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The nonzero terms of the cogrowth sequence of (Z/5Z)^*2 = <x|x^5=1> * <y|y^5=1> with respect to the generating set {(x,1), (1,y)}.
1

%I #23 Apr 05 2024 13:07:18

%S 1,2,12,92,792,7302,70464,702536,7178568,74771570,790906012,

%T 8472417384,91724327928,1001987961834,11030476949952,122247789508992,

%U 1362840516623944,15272530735735338,171946029518128956,1943927810200670820,22059590401383177792,251183781609841838444

%N The nonzero terms of the cogrowth sequence of (Z/5Z)^*2 = <x|x^5=1> * <y|y^5=1> with respect to the generating set {(x,1), (1,y)}.

%C The number of paths on the Cayley graph of ((Z/5Z)^*2, {(x,1), (1,y)} which start and end at the identity.

%C a(n) is the number of words of length 5n over the alphabet {x,y} that reduce to the empty string upon iteratively removing factors of x^5 and y^5. For n=2, the a(2)=12 words of length 10 that reduce to the empty string are x^10, y^10, x^i y^5 x^(5-i), for i=1..5, and y^i x^5 y^(5-i), for i=1..5.

%H J. P. Bell and M. J. Mishna, <a href="http://arxiv.org/abs/1805.08118">On the Complexity of the Cogrowth Sequence</a>, arXiv:1805.08118 [math.CO], 2018-2019.

%F G.f.: A(x) satisfies 32*x*A(x)^5 - (A(x)-1)*(A(x)+1)^4 = 0.

%F a(n) satisfies the recurrence (2120000*(5*n+1))*(5*n+2)*(5*n+3)*(5*n+4)*a(n) + (250*(160980199*n^4 + 1129209134*n^3 + 2872721885*n^2 + 3155706646*n + 1267579560))*a(n+1) - (50*(109722203*n^4 + 959367613*n^3 + 3144281425*n^2 + 4572924587*n + 2485585548))*a(n+2) + (60*(4290021*n^4 + 51502996*n^3 + 243316306*n^2 + 532456081*n + 451079946))*a(n+3) - (3*(2673299*n^4 + 44756419*n^3 + 283571239*n^2 + 805783469*n + 866093430))*a(n+4) + (4008*(n+5))*(4*n+17)*(2*n+9)*(4*n+19)*a(n+5) = 0.

%F a(n) ~ 5^(5*n + 1/2) / (9 * sqrt(Pi) * n^(3/2) * 2^(8*n - 3/2)). - _Vaclav Kotesovec_, Oct 24 2023

%F a(n) = binomial(5*n,n) - 3 * Sum_{k=0..n-1} binomial(5*n,k). - _Seiichi Manyama_, Apr 05 2024

%t terms = 22;

%t A[_] = 0; Do[A[x_] = (1 + 4 A[x] + 6 A[x]^2 + 4 A[x]^3 + A[x]^4 + 32 x A[x]^5)/(1 + A[x])^4 + O[x]^terms // Normal, {terms}];

%t CoefficientList[A[x], x] (* _Jean-François Alcover_, Nov 16 2018 *)

%Y Related cogrowth sequences (Z/2Z)^*2: A126869; (Z/3Z)^*2: A047098; (Z/4Z)^*2: A107026.

%K nonn

%O 0,2

%A _Marni Mishna_, May 22 2018