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A363305 Expansion of g.f. A(x) satisfying A(x) = 1 + x*(A(x)^5 + A(x)^9).

%I A363305 #10 Jul 13 2023 08:36:36
%S A363305 1,2,28,576,13968,371280,10465152,307252032,9295409664,287758274304,
%T A363305 9071667965184,290237226038272,9399819302979584,307570021821937664,
%U A363305 10152439243763290112,337658352835320934400,11304320019217804476416,380650592731460987617280
%N A363305 Expansion of g.f. A(x) satisfying A(x) = 1 + x*(A(x)^5 + A(x)^9).
%H A363305 Seiichi Manyama, <a href="/A363305/b363305.txt">Table of n, a(n) for n = 0..500</a>
%F A363305 G.f.: A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
%F A363305 (1) A(x) = 1 + x*(A(x)^5 + A(x)^9).
%F A363305 (2) a(n) = Sum_{k=0..n} binomial(n, k)*binomial(5*n+4*k+1, n)/(5*n+4*k+1) for n >= 0.
%e A363305 G.f.: A(x) = 1 + 2*x + 28*x^2 + 576*x^3 + 13968*x^4 + 371280*x^5 + 10465152*x^6 + 307252032*x^7 + 9295409664*x^8 + ...
%e A363305 where A(x) = 1 + x*(A(x)^5 + A(x)^9).
%e A363305 RELATED SERIES.
%e A363305 A(x)^5 = 1 + 10*x + 180*x^2 + 4080*x^3 + 104160*x^4 + 2858352*x^5 + 82336320*x^6 + 2455727040*x^7 + ...
%e A363305 A(x)^9 = 1 + 18*x + 396*x^2 + 9888*x^3 + 267120*x^4 + 7606800*x^5 + 224915712*x^6 + 6839682624*x^7 + ...
%o A363305 (PARI) {a(n) = sum(k=0, n, binomial(n, k)*binomial(5*n+4*k+1, n)/(5*n+4*k+1) )}
%o A363305 for(n=0, 20, print1(a(n), ", "))
%Y A363305 Cf. A027307, A363311, A363304.
%K A363305 nonn
%O A363305 0,2
%A A363305 _Paul D. Hanna_, May 29 2023