A346672 a(n) = Sum_{k=0..n} binomial(8*k,k) / (7*k + 1).

1, 2, 10, 102, 1342, 19620, 305004, 4943352, 82595376, 1412486081, 24602515801, 434935956337, 7783978950825, 140752989839105, 2567623696254905, 47195200645619009, 873239636055018809, 16251426606785706209, 304007720310330530081, 5713101394865420846381

Offset: 0

Author

Ilya Gutkovskiy, Jul 28 2021

Keywords

• nonn

Comments

Partial sums of A007556.

In general, for m > 1, Sum_{k=0..n} binomial(m*k,k) / ((m-1)*k + 1) ~ m^(m*(n+1) + 1/2) / (sqrt(2*Pi) * (m^m - (m-1)^(m-1)) * n^(3/2) * (m-1)^((m-1)*n + 3/2)). - Vaclav Kotesovec, Jul 30 2021

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..768

Crossrefs

Cf. A007556, A014137, A104859, A345367, A345368, A346065, A346650, A346671.

Programs

Mathematica

Table[Sum[Binomial[8 k, k]/(7 k + 1), {k, 0, n}], {n, 0, 19}]
nmax = 19; A[] = 0; Do[A[x] = 1/(1 - x) + x (1 - x)^7 A[x]^8 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

PARI

a(n) = sum(k=0, n, binomial(8*k, k)/(7*k+1)); \\ Michel Marcus, Jul 28 2021

Formula

G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x * (1 - x)^7 * A(x)^8.

a(n) ~ 2^(24*n + 25) / (15953673 * sqrt(Pi) * n^(3/2) * 7^(7*n + 3/2)). - Vaclav Kotesovec, Jul 30 2021