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%I #13 Nov 20 2021 07:18:40
%S 1,0,6,51,578,7011,89931,1198798,16445122,230643888,3292247673,
%T 47672499727,698569117499,10339672571689,154357100458366,
%U 2321475460350492,35140713973159266,534971413383669580,8185501429052369700,125811555778930237392,1941590759206061655069
%N a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(7*k,k) / (6*k + 1).
%C Inverse binomial transform of A002296.
%H Seiichi Manyama, <a href="/A346667/b346667.txt">Table of n, a(n) for n = 0..500</a>
%F G.f. A(x) satisfies: A(x) = 1 / (1 + x) + x * (1 + x)^5 * A(x)^7.
%F G.f.: Sum_{k>=0} ( binomial(7*k,k) / (6*k + 1) ) * x^k / (1 + x)^(k+1).
%F a(n) ~ 776887^(n + 3/2) / (282475249 * 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[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)^5 A[x]^7 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t nmax = 20; CoefficientList[Series[Sum[(Binomial[7 k, k]/(6 k + 1)) x^k/(1 + x)^(k + 1), {k, 0, nmax}], {x, 0, nmax}], x]
%t Table[(-1)^n HypergeometricPFQ[{1/7, 2/7, 3/7, 4/7, 5/7, 6/7, -n}, {1/3, 1/2, 2/3, 5/6, 1, 7/6}, 823543/46656], {n, 0, 20}]
%o (PARI) a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n,k)*binomial(7*k,k)/(6*k + 1)); \\ _Michel Marcus_, Jul 28 2021
%Y Cf. A002296, A005043, A346628, A346649, A346664, A346665, A346666, A346668.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jul 27 2021
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