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
Crossrefs
Cf. A186335.
Programs
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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
Comments