%I #13 Nov 15 2021 08:56:38

%S 1,2,16,212,3320,57024,1038928,19718512,385668448,7718866880,

%T 157326086656,3254310606208,68142850580480,1441588339943168,

%U 30765576147680000,661561298256228096,14319744815795062272,311756656998135770112,6822215641015820419072

%N G.f. A(x) satisfies: A(x) = (1 + x * A(x)^7) / (1 - x).

%H Seiichi Manyama, <a href="/A349313/b349313.txt">Table of n, a(n) for n = 0..500</a>

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

%F a(n) = F([(1+n)/6, (2+n)/6, (3+n)/6, (4+n)/6, (5+n)/6, 1+n/6, -n], [1/3, 1/2, 2/3, 5/6, 1, 7/6], -1), where F is the generalized hypergeometric function. - _Stefano Spezia_, Nov 14 2021

%F a(n) ~ sqrt(1 + 6*r) / (2 * 7^(2/3) * sqrt(3*Pi) * (1-r)^(1/3) * n^(3/2) * r^(n + 1/6)), where r = 0.04196526794785323647696104132939153750367778616407409162... is the real root of the equation 6^6 * (1-r)^7 = 7^7 * r. - _Vaclav Kotesovec_, Nov 15 2021

%t nmax = 18; A[_] = 0; Do[A[x_] = (1 + x A[x]^7)/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

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

%Y Cf. A002296, A006318, A346626, A346649, A349310, A349311, A349312, A349314.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Nov 14 2021