A349584 G.f. A(x) satisfies: A(x) = 1 / (1 - 2*x) + x * (1 - 2*x)^4 * A(x)^6.

1, 3, 14, 107, 1106, 13173, 168820, 2264298, 31356818, 444803666, 6429510234, 94356870748, 1402149248128, 21055387206719, 319007902203196, 4870481885025752, 74858763620576738, 1157339247553310574, 17985974981514604660, 280813589679135551721

Offset: 0

Author

Ilya Gutkovskiy, Nov 22 2021

Keywords

• nonn

Comments

Second binomial transform of A002295.

Crossrefs

Cf. A002295, A064613, A346648, A346762, A349581, A349582, A349590, A349591.

Programs

Mathematica

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]
Table[Sum[Binomial[n, k] Binomial[6 k, k] 2^(n - k)/(5 k + 1), {k, 0, n}], {n, 0, 19}]

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

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(6*k,k) * 2^(n-k) / (5*k+1).

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

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