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%I #13 Nov 20 2021 07:19:49
%S 1,1,6,57,629,7589,96942,1288729,17643920,247089010,3522891561,
%T 50964747400,746241617226,11038241689188,164696773030055,
%U 2475832560808858,37462189433509758,570112127356828846,8720472842436039280,133997057207982607092,2067402314984991892461
%N G.f. A(x) satisfies: A(x) = 1 + x * A(x)^7 / (1 + x).
%H Seiichi Manyama, <a href="/A349363/b349363.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(7*k,k) / (6*k+1).
%F a(n) = (-1)^(n+1)* F([8/7, 9/7, 10/7, 11/7, 12/7, 13/7, 1-n], [4/3, 3/2, 5/3, 11/6, 2, 13/6], 7^7/6^6), where F is the generalized hypergeometric function. - _Stefano Spezia_, Nov 15 2021
%F a(n) ~ 776887^(n + 1/2) / (343 * sqrt(Pi) * n^(3/2) * 2^(6*n + 2) * 3^(6*n + 3/2)). - _Vaclav Kotesovec_, Nov 17 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 = 20; 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[(-1)^(n - k) Binomial[n - 1, k - 1] Binomial[7 k, k]/(6 k + 1), {k, 0, n}], {n, 0, 20}]
%Y Cf. A001006, A002296, A127897, A317133, A346667, A346671 (binomial transform), A349334, A349361, A349362, A349364.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Nov 15 2021
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