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%I #16 Aug 07 2023 08:31:40
%S 1,1,8,85,1051,14197,203064,3022909,46347534,726894786,11606936525,
%T 188060979332,3084087347910,51094209834068,853859480938095,
%U 14376597494941454,243649099741045190,4153091242153905838,71152973167920086796,1224593757045581062444
%N G.f. A(x) satisfies A(x) = 1 + x * A(x)^7 / (1 - x).
%H Seiichi Manyama, <a href="/A349334/b349334.txt">Table of n, a(n) for n = 0..791</a>
%F a(n) = Sum_{k=0..n} binomial(n-1,k-1) * binomial(7*k,k) / (6*k+1).
%F a(n) ~ 870199^(n + 1/2) / (343 * sqrt(Pi) * n^(3/2) * 2^(6*n + 2) * 3^(6*n + 3/2)). - _Vaclav Kotesovec_, Nov 15 2021
%p a:= n-> coeff(series(RootOf(1+x*A^7/(1-x)-A, A), x, n+1), x, n):
%p seq(a(n), n=0..20); # _Alois P. Heinz_, Nov 15 2021
%t nmax = 19; 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 - 1, k - 1] Binomial[7 k, k]/(6 k + 1), {k, 0, n}], {n, 0, 19}]
%Y Cf. A002212, A307678, A349331, A349332, A349333, A349335.
%Y Cf. A002296, A346649 (partial sums), A349363.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Nov 15 2021
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