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%I #15 Nov 26 2021 05:01:05
%S 1,3,14,107,1106,13173,168820,2264298,31356818,444803666,6429510234,
%T 94356870748,1402149248128,21055387206719,319007902203196,
%U 4870481885025752,74858763620576738,1157339247553310574,17985974981514604660,280813589679135551721
%N G.f. A(x) satisfies: A(x) = 1 / (1 - 2*x) + x * (1 - 2*x)^4 * A(x)^6.
%C Second binomial transform of A002295.
%F a(n) = Sum_{k=0..n} binomial(n,k) * binomial(6*k,k) * 2^(n-k) / (5*k+1).
%F a(n) = 2^n*F([1/6, 1/3, 1/2, 2/3, 5/6, -n], [2/5, 3/5, 4/5, 1, 6/5], -3^6*(2/5)^5), where F is the generalized hypergeometric function. - _Stefano Spezia_, Nov 22 2021
%F a(n) ~ 2^(n - 15/2) * 26453^(n + 3/2) / (6561 * sqrt(3*Pi) * n^(3/2) * 5^(5*n + 3/2)). - _Vaclav Kotesovec_, Nov 26 2021
%t nmax = 19; A[_] = 0; Do[A[x_] = 1/(1 - 2 x) + x (1 - 2 x)^4 A[x]^6 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t Table[Sum[Binomial[n, k] Binomial[6 k, k] 2^(n - k)/(5 k + 1), {k, 0, n}], {n, 0, 19}]
%o (PARI) a(n) = sum(k=0, n, binomial(n,k)*binomial(6*k,k)*2^(n-k)/(5*k+1)); \\ _Michel Marcus_, Nov 23 2021
%Y Cf. A002295, A064613, A346648, A346762, A349581, A349582, A349590, A349591.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Nov 22 2021
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