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%I #14 Nov 26 2021 05:02:11
%S 1,3,15,132,1595,22134,329718,5136028,82579819,1359902823,22818697128,
%T 388728802702,6705324823466,116878939752376,2055505806198352,
%U 36427660285955808,649894104351874395,11662729497015257677,210383830525447606431,3812719304673511150854
%N G.f. A(x) satisfies: A(x) = 1 / (1 - 2*x) + x * (1 - 2*x)^5 * A(x)^7.
%C Second binomial transform of A002296.
%F a(n) = Sum_{k=0..n} binomial(n,k) * binomial(7*k,k) * 2^(n-k) / (6*k+1).
%F a(n) = 2^n*F([1/7, 2/7, 3/7, 4/7, 5/7, 6/7, -n], [1/3, 1/2, 2/3, 5/6, 1, 7/6], -7^7/(2^7*3^6)), where F is the generalized hypergeometric function. - _Stefano Spezia_, Nov 22 2021
%F a(n) ~ 916855^(n + 3/2) / (282475249 * sqrt(Pi) * n^(3/2) * 3^(6*n + 3/2) * 4^(3*n + 1)). - _Vaclav Kotesovec_, Nov 26 2021
%t nmax = 19; A[_] = 0; Do[A[x_] = 1/(1 - 2 x) + x (1 - 2 x)^5 A[x]^7 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t Table[Sum[Binomial[n, k] Binomial[7 k, k] 2^(n - k)/(6 k + 1), {k, 0, n}], {n, 0, 19}]
%o (PARI) a(n) = sum(k=0, n, binomial(n,k)*binomial(7*k,k)*2^(n-k)/(6*k+1)); \\ _Michel Marcus_, Nov 23 2021
%Y Cf. A002296, A064613, A346649, A346762, A349581, A349582, A349584, A349591.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Nov 22 2021