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 A349291 #18 Jul 10 2025 22:40:51
%S A349291 1,2,13,139,1775,24886,370099,5733304,91518691,1494815215,24862931821,
%T A349291 419674102147,7170713484877,123783319369420,2155542171446485,
%U A349291 37820343323942566,667957770644685811,11865421405897931581,211856917750711562695,3800040255017879663415
%N A349291 G.f. A(x) satisfies A(x) = 1 / ((1 - x) * (1 - x * A(x)^5)).
%H A349291 Seiichi Manyama, <a href="/A349291/b349291.txt">Table of n, a(n) for n = 0..500</a>
%F A349291 a(n) = Sum_{k=0..n} binomial(n+4*k,5*k) * binomial(6*k,k) / (5*k+1).
%F A349291 a(n) ~ sqrt(1 + 4*r) / (2^(6/5) * 3^(7/10) * sqrt(5*Pi*(1-r)) * n^(3/2) * r^(n + 1/5)), where r = 0.051436794119208432185504972091697516647... is the real root of the equation 6^6 * r = 5^5 * (1-r)^5. - _Vaclav Kotesovec_, Nov 14 2021
%F A349291 a(n) = 1 + Sum_{x_1, x_2, ..., x_6>=0 and x_1+x_2+...+x_6=n-1} Product_{k=1..6} a(x_k). - _Seiichi Manyama_, Jul 10 2025
%t A349291 nmax = 19; A[_] = 0; Do[A[x_] = 1/((1 - x) (1 - x A[x]^5)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t A349291 Table[Sum[Binomial[n + 4 k, 5 k] Binomial[6 k, k]/(5 k + 1), {k, 0, n}], {n, 0, 19}]
%Y A349291 Cf. A002295, A007317, A199475, A346648, A349289, A349290, A349292, A349293.
%K A349291 nonn
%O A349291 0,2
%A A349291 _Ilya Gutkovskiy_, Nov 13 2021