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A346668 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(8*k,k) / (7*k + 1).

%I A346668 #14 Nov 19 2021 09:19:02
%S A346668 1,0,7,70,917,12922,192591,2984156,47594289,776184997,12884436285,
%T A346668 216981375849,3698021707457,63663537870121,1105474964523293,
%U A346668 19339098305850757,340519405008643561,6030158137055187758,107328892461895007043,1918980244360791943044,34450128513971163342013
%N A346668 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(8*k,k) / (7*k + 1).
%C A346668 Inverse binomial transform of A007556.
%C A346668 In general, for m > 1, Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(m*k,k) / ((m-1)*k + 1) ~ (m^m - (m-1)^(m-1))^(n + 3/2)  / (sqrt(2*Pi) * m^((3*m-1)/2) * n^(3/2) * (m-1)^((m-1)*n + 3/2)). - _Vaclav Kotesovec_, Jul 30 2021
%H A346668 Seiichi Manyama, <a href="/A346668/b346668.txt">Table of n, a(n) for n = 0..500</a>
%F A346668 G.f. A(x) satisfies: A(x) = 1 / (1 + x) + x * (1 + x)^6 * A(x)^8.
%F A346668 G.f.: Sum_{k>=0} ( binomial(8*k,k) / (7*k + 1) ) * x^k / (1 + x)^(k+1).
%F A346668 a(n) ~ 15953673^(n + 3/2) / (34359738368 * sqrt(Pi) * n^(3/2) * 7^(7*n + 3/2)). - _Vaclav Kotesovec_, Jul 30 2021
%t A346668 Table[Sum[(-1)^(n - k) Binomial[n, k] Binomial[8 k, k]/(7 k + 1), {k, 0, n}], {n, 0, 20}]
%t A346668 nmax = 20; A[_] = 0; Do[A[x_] = 1/(1 + x) + x (1 + x)^6 A[x]^8 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t A346668 nmax = 20; CoefficientList[Series[Sum[(Binomial[8 k, k]/(7 k + 1)) x^k/(1 + x)^(k + 1), {k, 0, nmax}], {x, 0, nmax}], x]
%t A346668 Table[(-1)^n HypergeometricPFQ[{1/8, 1/4, 3/8, 1/2, 5/8, 3/4, 7/8, -n}, {2/7, 3/7, 4/7, 5/7, 6/7, 1, 8/7}, 16777216/823543], {n, 0, 20}]
%o A346668 (PARI) a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n,k)*binomial(8*k,k)/(7*k + 1)); \\ _Michel Marcus_, Jul 28 2021
%Y A346668 Cf. A005043, A007556, A346628, A346650, A346664, A346665, A346666, A346667.
%K A346668 nonn
%O A346668 0,3
%A A346668 _Ilya Gutkovskiy_, Jul 27 2021