A104979 Semidiagonal sums of triangle A104978: a(n) = Sum_{k=0..[n/2]} A104978(n-k,n-2*k).

1, 1, 4, 17, 85, 459, 2614, 15454, 93947, 583568, 3687761, 23633072, 153227250, 1003281314, 6624658716, 44062205158, 294938814921, 1985330061570, 13430612284606, 91262392343333, 622624395706714, 4263163419492661

Offset: 0

Author

Paul D. Hanna, Mar 30 2005

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A104978.

Programs

Magma

A104979:= func< n | (&+[Binomial(3*n-4*k, 3*n-5*k)*Binomial(3*n-5*k, n-2*k)/(2*n-3*k+1): k in [0..Floor(n/2)]]) >;
[A104979(n): n in [0..30]]; // G. C. Greubel, Jun 08 2021

Mathematica

Table[Sum[Binomial[3*n-4*k, 3*n-5*k]*Binomial[3*n-5*k, n-2*k]/(2*n-3*k+1), {k,0,n/2}], {n,0,30}] (* G. C. Greubel, Jun 08 2021 *)

PARI

{a(n)=sum(k=0,n\2, binomial(3*n-4*k,3*n-5*k)*binomial(3*n-5*k,n-2*k)/(2*n-3*k+1))}

Sage

def A104979(n): return sum( binomial(3*n-4*k, 3*n-5*k)*binomial(3*n-5*k, n-2*k)/(2*n-3*k+1) for k in (0..n//2) )
[A104979(n) for n in (0..30)] # G. C. Greubel, Jun 08 2021

Formula

a(n) = Sum_{k=0..floor(n/2)} A104978(n-k, n-2*k).

a(n) = Sum_{k=0..floor(n/2)} binomial(3*n-4*k, 3*n-5*k)*binomial(3*n-5*k, n-2*k)/(2*n-3*k+1).

G.f. satisfies: A(x) = 1 + x*A(x)^3 + x^2*A(x)^2. - Paul D. Hanna, May 27 2010