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 A346649 #12 Nov 15 2021 08:57:08
%S A346649 1,2,10,95,1146,15343,218407,3241316,49588850,776483636,12383420161,
%T A346649 200444399493,3284531747403,54378741581471,908238222519566,
%U A346649 15284835717461020,258933935458506210,4412025177612412048,75564998345532498844,1300158755391113561288
%N A346649 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(7*k,k) / (6*k + 1).
%C A346649 Binomial transform of A002296.
%H A346649 Seiichi Manyama, <a href="/A346649/b346649.txt">Table of n, a(n) for n = 0..500</a>
%F A346649 G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x * (1 - x)^5 * A(x)^7.
%F A346649 G.f.: Sum_{k>=0} ( binomial(7*k,k) / (6*k + 1) ) * x^k / (1 - x)^(k+1).
%F A346649 a(n) ~ 870199^(n + 3/2) / (282475249 * sqrt(Pi) * n^(3/2) * 2^(6*n + 2) * 3^(6*n + 3/2)). - _Vaclav Kotesovec_, Jul 30 2021
%t A346649 Table[Sum[Binomial[n, k] Binomial[7 k, k]/(6 k + 1), {k, 0, n}], {n, 0, 19}]
%t A346649 nmax = 19; A[_] = 0; Do[A[x_] = 1/(1 - x) + x (1 - x)^5 A[x]^7 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t A346649 nmax = 19; CoefficientList[Series[Sum[(Binomial[7 k, k]/(6 k + 1)) x^k/(1 - x)^(k + 1), {k, 0, nmax}], {x, 0, nmax}], x]
%t A346649 Table[HypergeometricPFQ[{1/7, 2/7, 3/7, 4/7, 5/7, 6/7, -n}, {1/3, 1/2, 2/3, 5/6, 1, 7/6}, -823543/46656], {n, 0, 19}]
%o A346649 (PARI) a(n) = sum(k=0, n, binomial(n,k)*binomial(7*k,k)/(6*k + 1)); \\ _Michel Marcus_, Jul 26 2021
%Y A346649 Cf. A002296, A007317, A188687, A346646, A346647, A346648, A346650.
%K A346649 nonn
%O A346649 0,2
%A A346649 _Ilya Gutkovskiy_, Jul 26 2021