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G.f. A(x) satisfies: A(x) = 1 / ((1 - x) * (1 - x * A(x)^6)).
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%I #11 Nov 15 2021 08:56:58

%S 1,2,15,190,2871,47643,838888,15389452,290951545,5629024955,

%T 110908062511,2217739684483,44891645810124,918086053852234,

%U 18941156419798530,393742848618632760,8239112912485293357,173406208518520952066,3668419587671991125142

%N G.f. A(x) satisfies: A(x) = 1 / ((1 - x) * (1 - x * A(x)^6)).

%H Seiichi Manyama, <a href="/A349292/b349292.txt">Table of n, a(n) for n = 0..500</a>

%F a(n) = Sum_{k=0..n} binomial(n+5*k,6*k) * binomial(7*k,k) / (6*k+1).

%F a(n) ~ sqrt(1 + 5*r) / (2 * 7^(2/3) * sqrt(3*Pi*(1-r)) * n^(3/2) * r^(n + 1/6)), where r = 0.043408935906208378827553096713877784793679356... is the root of the equation 7^7 * r = 6^6 * (1-r)^6. - _Vaclav Kotesovec_, Nov 14 2021

%t nmax = 18; A[_] = 0; Do[A[x_] = 1/((1 - x) (1 - x A[x]^6)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

%t Table[Sum[Binomial[n + 5 k, 6 k] Binomial[7 k, k]/(6 k + 1), {k, 0, n}], {n, 0, 18}]

%Y Cf. A002296, A007317, A199475, A346649, A349289, A349290, A349291, A349293.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Nov 13 2021