A186335 A transform of the central binomial coefficients.

1, 1, 4, 7, 21, 46, 127, 309, 832, 2131, 5709, 15010, 40281, 107423, 289314, 778087, 2103361, 5687938, 15427099, 41880357, 113912236, 310148223, 845598389, 2307657222, 6304306171, 17237338021, 47170965082, 129181447969, 354027263457, 970851960736, 2664008539017

Offset: 0

Author

Paul Barry, Feb 18 2011

Keywords

• nonn
• easy

Comments

Hankel transform is (-1)^n*A128056(n).

Programs

Maple

A186335 := proc(n)
    add(add(binomial(k-j,n-k-j)*binomial(k,j)*A000984(n-k-j),j=0..n),k=0..n) ;
end proc: # R. J. Mathar, Feb 13 2015

Mathematica

Table[Sum[Sum[Binomial[k-j,n-k-j]*Binomial[k,j]*Binomial[2*(n-k-j),n-k-j], {j,0,n}], {k,0,n}], {n,0,40}] (* 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)*A000984(n-k-j)}}.

Conjecture: n*a(n) +(-2*n+1)*a(n-1) +5*(-n+1)*a(n-2) +3*(2*n-3)*a(n-3) +5*(n-2)*a(n-4)=0. - R. J. Mathar, Feb 13 2015

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