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%I #13 Nov 20 2021 07:20:00
%S 1,1,7,77,987,13839,205513,3176747,50578445,823779286,13660621282,
%T 229865812134,3915003083306,67361559577578,1169138502393414,
%U 20444573270374050,359858503314494318,6370677542063831319,113359050598950194801,2026309136822686950087
%N G.f. A(x) satisfies: A(x) = 1 + x * A(x)^8 / (1 + x).
%C In general, for m > 1, Sum_{k=0..n} (-1)^(n-k) * binomial(n-1,k-1) * binomial(m*k,k) / ((m-1)*k+1) ~ (m-1)^(m/2 - 2) * (m^m/(m-1)^(m-1) - 1)^(n + 1/2) / (sqrt(2*Pi) * m^((m-1)/2) * n^(3/2)). - _Vaclav Kotesovec_, Nov 17 2021
%H Seiichi Manyama, <a href="/A349364/b349364.txt">Table of n, a(n) for n = 0..500</a>
%F a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n-1,k-1) * binomial(8*k,k) / (7*k+1).
%F a(n) = (-1)^(n+1)*F([9/8, 5/4, 11/8, 3/2, 13/8, 7/4, 15/8, 1-n], [9/7, 10/7, 11/7, 12/7, 13/7, 2, 15/7], 8^8/7^7), where F is the generalized hypergeometric function. - _Stefano Spezia_, Nov 15 2021
%F a(n) ~ 15953673^(n + 1/2) / (2048 * sqrt(Pi) * n^(3/2) * 7^(7*n + 3/2)). - _Vaclav Kotesovec_, Nov 17 2021
%p a:= n-> coeff(series(RootOf(1+x*A^8/(1+x)-A, A), x, n+1), x, n):
%p seq(a(n), n=0..19); # _Alois P. Heinz_, Nov 15 2021
%t nmax = 19; A[_] = 0; Do[A[x_] = 1 + x A[x]^8/(1 + x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t Table[Sum[(-1)^(n - k) Binomial[n - 1, k - 1] Binomial[8 k, k]/(7 k + 1), {k, 0, n}], {n, 0, 19}]
%Y Cf. A001006, A007556, A127897, A317133, A346668, A346672 (binomial transform), A349335, A349361, A349362, A349363.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Nov 15 2021
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