A277638 Binomial partial sums of sequence A007004.

1, 4, 37, 520, 8803, 165292, 3320023, 69943804, 1526981575, 34271322316, 786371041603, 18372739163632, 435772652437381, 10468369504009060, 254238148448959729, 6233226769739934964, 154092763036678601551, 3837301178450916902428, 96181503100227675085675

Offset: 0

Author

Emanuele Munarini, Oct 25 2016

Keywords

• nonn

Crossrefs

Cf. A007004.

Programs

Mathematica

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

Maxima

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

Formula

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

a(n) = hypergeometric(1/3,2/3,-n; 1,2; -27).

a(n) == 1 (mod 3) for all natural n.

E.g.f.: exp(t) * hypergeometric(1/3,2/3; 1,2; 27*t).

From Vaclav Kotesovec, Oct 26 2016: (Start)

Recurrence: n*(n+1)*a(n) = 2*(3*n-1)*(5*n-3)*a(n-1) - (n-1)*(57*n-56)*a(n-2) + 28*(n-2)*(n-1)*a(n-3).

a(n) ~ 2^(2*n+3) * 7^(n+2) / (3^(11/2) * Pi * n^2).

(End)

Diff. eq. satisfied by the ordinary g.f.: t*(1-t)^2*(1-28*t)*A''(t)+2*(1-t)*(1-2*t)*(1-28*t)*A'(t)-2*(4-29*t+28*t^2)*A(t)=0. - Emanuele Munarini, Oct 28 2016

G.f.: hypergeom([1/3, 2/3],[2],27*x/(1-x))/(1-x). - Mark van Hoeij, Nov 28 2024