A397541 a(n) = Sum_{k=0..n} binomial(2*n+3*k,k).

1, 6, 53, 531, 5628, 61548, 686630, 7766396, 88737998, 1021790013, 11837767158, 137825528185, 1611269752236, 18901744731931, 222387215999071, 2623111989210510, 31008557957020701, 367272410007660652, 4357537180835561768, 51779701659010388535, 616134542980865344208

Offset

0,2

Links

Table of n, a(n) for n=0..20.

Formula

G.f.: 1/((1-5*x*g^4) * (1-x*g^2)) where g = 1+x*g^5 is the g.f. of A002294.

G.f.: g^4/((5-4*g) * (1-g+g^3)) where g = 1+x*g^5 is the g.f. of A002294.

Here and below, binomial(N,k) = 0 for k<0.

This is the special case l=2, m=3, c=4, r=1, s=0 of the following family. For integers l, m, c in Z, and for any constants r and s, define a_{l,m,c,r,s}(n) = Sum_{k=0..n} (r*binomial(l*n+m*k-m-1+c,k) - s*binomial(l*n+m*k-m-1+c,k-1)). A_{l,m,c,r,s}(x) = Sum_{n>=0} a_{l,m,c,r,s}(n)*x^n = t^c*(r+s-s*t)/((l+m-(l+m-1)*t)*(1-t+t^m)), where t = t(x) satisfies t = 1 + x*t^(l+m), equivalently y = x*(1+y)^(l+m) with y=t-1.

Prog

(PARI) a(n) = sum(k=0, n, binomial(2*n+3*k, k));

Crossrefs

Cf. A045721, A390407.

Cf. A002294, A389239, A397538.

Sequence in context: A223345 A362907 A388130 * A369506 A066357 A276365

Adjacent sequences: A397538 A397539 A397540 * A397542 A397543 A397544

Keyword

nonn,easy

Author

Seiichi Manyama, Jun 30 2026

Status

approved