A397537 a(n) = Sum_{k=0..n} binomial(n+5*k-6,k).

1, 1, 17, 244, 3378, 46785, 652256, 9156348, 129320648, 1835951015, 26179257723, 374690184757, 5379908547086, 77459905683802, 1117955506225531, 16169269942413750, 234298085780246524, 3400717317239096007, 49433392409740652030, 719538498483347997844, 10486149676767529103732

Offset

0,3

Links

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

Formula

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

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

Crossrefs

Cf. A088218, A397509, A397535, A397536.

Cf. A002295.

Sequence in context: A196987 A051560 A259347 * A324358 A294608 A294810

Adjacent sequences: A397534 A397535 A397536 * A397538 A397539 A397540

Keyword

nonn,easy,new

Author

Seiichi Manyama, Jun 30 2026

Status

approved