A186334 A transform of the Catalan numbers.

1, 1, 3, 5, 12, 24, 56, 123, 291, 677, 1637, 3954, 9757, 24171, 60648, 152929, 388822, 993216, 2551808, 6582899, 17055507, 44341141, 115671498, 302627130, 793951897, 2088103609, 5504504961, 14541271283, 38489869502, 102066761622, 271122837895

Offset: 0

Author

Paul Barry, Feb 18 2011

Keywords

• nonn
• easy

Comments

Hankel transform is A094967(n+1) (F(2n+1) repeated).

Crossrefs

Cf. A186335.

Programs

Mathematica

Table[Sum[Sum[Binomial[k-j,n-k-j] * Binomial[k,j] * If[n-k-j>=0, CatalanNumber[n-k-j], 0], {j,0,n}], {k,0,n}], {n,0,30}] (* Vaclav Kotesovec, Oct 30 2017 *)

Formula

a(n)=sum{k=0..n, sum{j=0..n, binomial(k-j,n-k-j)*binomial(k,j)*if(n-k-j>=0, A000108(n-k-j),0)}}

Conjecture: (n+2)*a(n) +2*(-n-1)*a(n-1) +(-5*n+4)*a(n-2) +2*(3*n-4)*a(n-3) +5*(n-2)*a(n-4)=0. - R. J. Mathar, Nov 07 2014

a(n) ~ 21^(1/4) * ((1+sqrt(21))/2)^(n + 5/2) / (8 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Oct 30 2017