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
EXAMPLE
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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 23 2007
STATUS
approved