%I #10 Oct 30 2017 06:13:00

%S 1,1,3,5,12,24,56,123,291,677,1637,3954,9757,24171,60648,152929,

%T 388822,993216,2551808,6582899,17055507,44341141,115671498,302627130,

%U 793951897,2088103609,5504504961,14541271283,38489869502,102066761622,271122837895

%N A transform of the Catalan numbers.

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

%F 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)}}

%F 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

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

%t 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 *)

%Y Cf. A186335.

%K nonn,easy

%O 0,3

%A _Paul Barry_, Feb 18 2011