A397565 a(n) = Sum_{k=0..n} binomial(3*n+3*k-4,k).

1, 3, 34, 428, 5628, 75926, 1041876, 14471417, 202839843, 2863267846, 40646324515, 579673788822, 8298771433240, 119193890037793, 1716729717777120, 24785397074603618, 358595277154362550, 5197804355574056304, 75466114062838427516, 1097299074257879775625

Offset

0,2

Links

Table of n, a(n) for n=0..19.

Formula

G.f.: 1/(g^4 * (1-6*x*g^5) * (1-x*g^3)) where g = 1+x*g^6 is the g.f. of A002295.

G.f.: 1/((6-5*g) * (1-g+g^3)) 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=3, 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(3*n+3*k-4, k));

Crossrefs

Cf. A397512, A397537, A397539.

Cf. A002295.

Sequence in context: A045727 A105713 A397507 * A376137 A380491 A367840

Adjacent sequences: A397562 A397563 A397564 * A397567 A397568 A397574

Keyword

nonn,easy,new

Author

Seiichi Manyama, Jul 01 2026

Status

approved