A277637 Partial sums of A007004.

1, 4, 34, 454, 7384, 133510, 2583958, 52468078, 1104191608, 23891534758, 528527606698, 11905777228618, 272269758961018, 6306419621308618, 147677930682023818, 3491114167267938298, 83217317955857060788, 1998209467779811473538, 48293984598611551487038, 1174012306200941229710038, 28689784424223599507417938

Offset: 0

Author

Emanuele Munarini, Oct 25 2016

Keywords

• nonn

Crossrefs

Cf. A007004.

Programs

Mathematica

Table[Sum[Multinomial[k, k, k]/(k + 1), {k, 0, n}], {n, 0, 100}]

Maxima

makelist(sum(multinomial_coeff(k,k,k)/(k+1),k,0,n),n,0,12);

Formula

a(n) = Sum_{k=0..n} multinomial(k,k,k)/(k+1).

Recurrence: (n+2)*(n+3)*a(n+2)-(28*n^2+86*n+66)*a(n+1)+3*(3*n+5)*(3*n+4)*a(n)=0.

a(n) = hypergeometric(1/3,2/3;2;27) - (multinomial(n+1,n+1,n+1)/(n+2)) * hypergeometric(1,n+4/3,n+5/3;n+2,n+3;27).

a(n) = 0 (mod 2) and a(n) = 1 (mod 3), for all natural n.

G.f.: hypergeometric(1/3,2/3;2;27*t)/(1-t).

a(n) ~ 3^(3*n+7/2) / (52*Pi*n^2). - Vaclav Kotesovec, Oct 30 2016