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

Offset

0,2

Links

Table of n, a(n) for n=0..17.

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

Mathematica

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

Prog

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

Crossrefs

Cf. A007004.

Sequence in context: A316877 A349254 A365778 * A352237 A349714 A121080

Adjacent sequences: A277635 A277636 A277637 * A277639 A277640 A277641

Keyword

nonn

Author

Emanuele Munarini, Oct 25 2016

Status

approved