A397535 a(n) = Sum_{k=0..n} binomial(n+3*k-4,k).

1, 1, 8, 69, 598, 5229, 46148, 410564, 3677205, 33118004, 299653447, 2721863556, 24805834384, 226714986482, 2077219038583, 19073322225093, 175469528064836, 1617021669150125, 14924185776171959, 137930294999776878, 1276338562136112694, 11823881876462306463

Offset

0,3

Links

Table of n, a(n) for n=0..21.

Formula

G.f.: 1/(g^4 * (1-4*x*g^3) * (1-x*g)) where g = 1+x*g^4 is the g.f. of A002293.

G.f.: 1/((4-3*g) * (1-g+g^3)) where g = 1+x*g^4 is the g.f. of A002293.

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

This is the special case l=1, m=3, 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(n+3*k-4, k));

Crossrefs

Cf. A389239, A390383, A390412, A390471.

Cf. A088218, A397509, A397536, A397537.

Cf. A002293.

Sequence in context: A152279 A015575 A214344 * A228421 A152109 A390744

Adjacent sequences: A397532 A397533 A397534 * A397536 A397537 A397538

Keyword

nonn,easy,new

Author

Seiichi Manyama, Jun 30 2026

Status

approved