This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A349302 #14 Jun 06 2024 08:21:32
%S A349302 1,0,1,6,43,321,2500,20096,165621,1392397,11896823,103014141,
%T A349302 902035660,7974080834,71070247438,637937825112,5761970031357,
%U A349302 52329993278856,477588786637264,4377832437503643,40288077072190109,372086539388626537,3447632819399550915
%N A349302 G.f. A(x) satisfies: A(x) = 1 / ((1 + x) * (1 - x * A(x)^6)).
%H A349302 Seiichi Manyama, <a href="/A349302/b349302.txt">Table of n, a(n) for n = 0..500</a>
%F A349302 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n+5*k,6*k) * binomial(7*k,k) / (6*k+1).
%F A349302 a(n) ~ sqrt(1 - 5*r) / (2 * 7^(2/3) * sqrt(3*Pi*(1+r)) * n^(3/2) * r^(n + 1/6)), where r = 0.1008057775745727124639860500770912830001828593281202101426766... is the root of the equation 7^7 * r = 6^6 * (1+r)^6. - _Vaclav Kotesovec_, Nov 14 2021
%F A349302 From _Peter Bala_, Jun 02 2024: (Start)
%F A349302 A(x) = 1/(1 + x)*F(x/(1 + x)^6), where F(x) = Sum_{n >= 0} A002296(n)*x^n.
%F A349302 A(x) = 1/(1 + x) + x*A(x)^7. (End)
%t A349302 nmax = 22; 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 A349302 Table[Sum[(-1)^(n - k) Binomial[n + 5 k, 6 k] Binomial[7 k, k]/(6 k + 1), {k, 0, n}], {n, 0, 22}]
%Y A349302 Cf. A002296, A005043, A346627, A346667, A349292, A349299, A349300, A349301, A349303.
%K A349302 nonn,easy
%O A349302 0,4
%A A349302 _Ilya Gutkovskiy_, Nov 13 2021