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%I #14 Jul 30 2021 10:18:58
%S 1,0,8,84,1156,17122,268262,4370086,73281938,1256608767,21933420953,
%T 388400019583,6960642974905,126008367913375,2300862338502425,
%U 42326714610861679,783717720798538121,14594469249932149279,273161824453612674593,5135931850101477641707
%N a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(8*k,k) / (7*k + 1).
%C In general, for m > 1, Sum_{k=0..n} (-1)^(n-k) * binomial(m*k,k) / ((m-1)*k + 1) ~ m^(m*(n+1) + 1/2) / (sqrt(2*Pi) * (m^m + (m-1)^(m-1)) * n^(3/2) * (m-1)^((m-1)*n + 3/2)). - _Vaclav Kotesovec_, Jul 30 2021
%H Seiichi Manyama, <a href="/A346684/b346684.txt">Table of n, a(n) for n = 0..768</a>
%F G.f. A(x) satisfies: A(x) = 1 / (1 + x) + x * (1 + x)^7 * A(x)^8.
%F a(n) ~ 2^(24*n + 25) / (17600759 * sqrt(Pi) * n^(3/2) * 7^(7*n + 3/2)). - _Vaclav Kotesovec_, Jul 30 2021
%t Table[Sum[(-1)^(n - k) Binomial[8 k, k]/(7 k + 1), {k, 0, n}], {n, 0, 19}]
%t nmax = 19; A[_] = 0; Do[A[x_] = 1/(1 + x) + x (1 + x)^7 A[x]^8 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%o (PARI) a(n) = sum(k=0, n, (-1)^(n-k)*binomial(8*k, k)/(7*k + 1)); \\ _Michel Marcus_, Jul 29 2021
%Y Cf. A007556, A032357, A188678, A346668, A346672.
%Y Cf. A346680, A346681, A346682, A346683.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jul 29 2021
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