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%I #13 Jul 30 2021 10:19:33
%S 1,0,7,63,756,9716,132062,1865626,27124049,403197584,6100155272,
%T 93626517858,1454221328232,22815183746508,361030984965596,
%U 5755543515895284,92350704790963431,1490287557170676816,24171116970619575559,393808998160695560841,6442255541764422795759
%N a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(7*k,k) / (6*k + 1).
%H Seiichi Manyama, <a href="/A346683/b346683.txt">Table of n, a(n) for n = 0..806</a>
%F G.f. A(x) satisfies: A(x) = 1 / (1 + x) + x * (1 + x)^6 * A(x)^7.
%F a(n) ~ 7^(7*n + 15/2) / (870199 * sqrt(Pi) * n^(3/2) * 2^(6*n + 2) * 3^(6*n + 3/2)). - _Vaclav Kotesovec_, Jul 30 2021
%t Table[Sum[(-1)^(n - k) Binomial[7 k, k]/(6 k + 1), {k, 0, n}], {n, 0, 20}]
%t nmax = 20; A[_] = 0; Do[A[x_] = 1/(1 + x) + x (1 + x)^6 A[x]^7 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%o (PARI) a(n) = sum(k=0, n, (-1)^(n-k)*binomial(7*k, k)/(6*k + 1)); \\ _Michel Marcus_, Jul 29 2021
%Y Cf. A002296, A032357, A188678, A346667, A346671.
%Y Cf. A346680, A346681, A346682, A346684.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jul 29 2021
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