A346065 a(n) = Sum_{k=0..n} binomial(6*k,k) / (5*k + 1).

1, 2, 8, 59, 565, 6046, 68878, 818276, 10021910, 125629220, 1603943486, 20783993414, 272641113110, 3613484662965, 48313969712685, 650888627139801, 8826840286257595, 120398870546499685, 1650711840886884265, 22735860619151166130, 314441081323870331656

Offset: 0

Author

Ilya Gutkovskiy, Jul 28 2021

Keywords

• nonn

Comments

Partial sums of A002295.

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 28 2021

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..856

Crossrefs

Cf. A002295, A014137, A104859, A345367, A345368, A346648, A346671, A346672.

Programs

Mathematica

Table[Sum[Binomial[6 k, k]/(5 k + 1), {k, 0, n}], {n, 0, 20}]
nmax = 20; A[] = 0; Do[A[x] = 1/(1 - x) + x (1 - x)^5 A[x]^6 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

PARI

a(n) = sum(k=0, n, binomial(6*k, k)/(5*k+1)); \\ Michel Marcus, Jul 28 2021

Formula

G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x * (1 - x)^5 * A(x)^6.

a(n) ~ 2^(6*n + 6) * 3^(6*n + 13/2) / (43531 * sqrt(Pi) * n^(3/2) * 5^(5*n + 3/2)). - Vaclav Kotesovec, Jul 28 2021