A128079 a(n) = Sum_{k=0..n} A000984(k)*A001263(n+1,k+1), where A000984 is the central binomial coefficients and A001263 is the Narayana triangle.

1, 3, 13, 69, 411, 2633, 17739, 124029, 892327, 6567285, 49235715, 374841195, 2890994445, 22545855855, 177524073021, 1409591810133, 11275693221519, 90792020672429, 735367765159347, 5987665336600683, 48987680485918149

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

a(n) = Sum_{k=0..n} C(2k,k)*C(n,k)*C(n+1,k)/(k+1).

Recurrence: (n+1)*(n+2)*a(n) = (7*n^2+11*n+6)*a(n-1) + 3*(7*n^2-19*n+6)*a(n-2) - 27*(n-2)*(n-1)*a(n-3) . - Vaclav Kotesovec, Oct 20 2012

a(n) ~ 3^(2*n+7/2)/(8*Pi*n^2) . - Vaclav Kotesovec, Oct 20 2012

a(n) = ((n+3)^2*A005802(n+1)-(n-3)*(n+1)*A005802(n))/12. - Mark van Hoeij, Nov 12 2023

Example

Illustrate a(n) = Sum_{k=0..n} A000984(k)*A001263(n+1,k+1) by:
a(2) = 1*(1) + 2*(3) + 6*(1) = 13;
a(3) = 1*(1) + 2*(6) + 6*(6) + 20*(1) = 69;
a(4) = 1*(1) + 2*(10)+ 6*(20)+ 20*(10)+ 70*(1) = 411.
The Narayana triangle A001263(n+1,k+1) = C(n,k)*C(n+1,k)/(k+1) begins:
1;
1, 1;
1, 3, 1;
1, 6, 6, 1;
1, 10, 20, 10, 1;
1, 15, 50, 50, 15, 1; ...

Mathematica

Table[Sum[Binomial[2*k, k]*Binomial[n, k]*Binomial[n+1, k]/(k+1), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 20 2012 *)

Prog

(PARI) {a(n)=sum(k=0, n, binomial(2*k, k)*binomial(n, k)*binomial(n+1, k)/(k+1))}

Crossrefs

Cf. A000984, A001263.

Sequence in context: A020107 A284718 A284719 * A074534 A153395 A243688

Adjacent sequences: A128076 A128077 A128078 * A128080 A128081 A128082

Keyword

nonn

Author

Paul D. Hanna, Feb 23 2007

Status

approved