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

0,3

Comments

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

Links

Table of n, a(n) for n=0..30.

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

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 *)

Crossrefs

Cf. A186335.

Sequence in context: A027246 A185087 A090345 * A303587 A151524 A030270

Adjacent sequences: A186331 A186332 A186333 * A186335 A186336 A186337

Keyword

nonn,easy

Author

Paul Barry, Feb 18 2011

Status

approved