login
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
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
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
Sequence in context: A020107 A284718 A284719 * A074534 A153395 A243688
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 23 2007
STATUS
approved