A397539 a(n) = Sum_{k=0..n} binomial(2*n+4*k-5,k).

1, 2, 25, 328, 4394, 59883, 826952, 11535925, 162202029, 2295079168, 32641265307, 466210121163, 6682645676656, 96081057775690, 1385055494381483, 20011937622661306, 289722936844910404, 4201924638932790618, 61038153035664510382, 887914762634986995965, 12932943216967486046109

Offset

0,2

Links

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

Formula

G.f.: 1/(g^5 * (1-6*x*g^5) * (1-x*g^2)) where g = 1+x*g^6 is the g.f. of A002295.

G.f.: 1/((6-5*g) * (1-g+g^4)) where g = 1+x*g^6 is the g.f. of A002295.

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

This is the special case l=2, m=4, c=0, 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+4*k-5, k));

Crossrefs

Cf. A165817, A397510, A397538, A397540.

Cf. A002295.

Sequence in context: A270346 A367506 A387047 * A212022 A198710 A074209

Adjacent sequences: A397536 A397537 A397538 * A397540 A397541 A397542

Keyword

nonn,easy

Author

Seiichi Manyama, Jun 30 2026

Status

approved