A345368 a(n) = Sum_{k=0..n} binomial(5*k,k) / (4*k + 1).

1, 2, 7, 42, 327, 2857, 26608, 258488, 2588933, 26539288, 277082658, 2936050788, 31494394563, 341325970323, 3731742758203, 41108999917483, 455850863463768, 5084213586320193, 56997201842602368, 641906808539396253, 7258985455500009623, 82393287049581399283

Offset: 0

Author

Ilya Gutkovskiy, Jul 28 2021

Keywords

• nonn

Comments

Partial sums of A002294.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..925

Crossrefs

Cf. A002294, A014137, A104859, A345367, A346065, A346647, A346671, A346672.

Programs

Mathematica

Table[Sum[Binomial[5 k, k]/(4 k + 1), {k, 0, n}], {n, 0, 21}]
nmax = 21; A[] = 0; Do[A[x] = 1/(1 - x) + x (1 - x)^4 A[x]^5 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

PARI

a(n) = sum(k=0, n, binomial(5*k, k)/(4*k+1)); \\ Michel Marcus, Jul 28 2021

Formula

G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x * (1 - x)^4 * A(x)^5.

a(n) ~ 5^(5*n + 11/2) / (2869 * sqrt(Pi) * n^(3/2) * 2^(8*n + 7/2)). - Vaclav Kotesovec, Jul 28 2021