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%I #13 Nov 15 2021 08:56:43
%S 1,2,10,74,642,6082,60970,635818,6826690,74958914,837833482,
%T 9500939978,109037364930,1264049402754,14780619799722,174121322204074,
%U 2064572904600706,24620095821589378,295087003429677322,3552841638851183690,42950428996378731010
%N G.f. A(x) satisfies: A(x) = (1 + x * A(x)^4) / (1 - x).
%H Seiichi Manyama, <a href="/A349310/b349310.txt">Table of n, a(n) for n = 0..500</a>
%F a(n) = Sum_{k=0..n} binomial(n+3*k,4*k) * binomial(4*k,k) / (3*k+1).
%F a(n) = F([(1+n)/3, (2+n)/3, (3+n)/3, -n], [2/3, 1, 4/3], -1), where F is the generalized hypergeometric function. - _Stefano Spezia_, Nov 14 2021
%F a(n) ~ sqrt(1 + 3*r) / (2^(13/6) * sqrt(3*Pi) * (1-r)^(1/6) * n^(3/2) * r^(n + 1/3)), where r = 0.0766602099042102089064087954661556186872273232742446843... is the smallest real root of the equation 3^3 * (1-r)^4 = 4^4 * r. - _Vaclav Kotesovec_, Nov 15 2021
%t nmax = 20; A[_] = 0; Do[A[x_] = (1 + x A[x]^4)/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t Table[Sum[Binomial[n + 3 k, 4 k] Binomial[4 k, k]/(3 k + 1), {k, 0, n}], {n, 0, 20}]
%Y Cf. A002293, A006318, A346626, A346646, A349311, A349312, A349313, A349314.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Nov 14 2021
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