A397513 a(n) = Sum_{k=0..n} binomial(5*n+2*k-3,k).

1, 5, 65, 951, 14651, 232407, 3756831, 61534891, 1017777761, 16960532172, 284320206391, 4789320531464, 80998807949865, 1374509101940902, 23391969665137355, 399085386227367204, 6823496869562499259, 116890000259795561005, 2005779247640013424776

Offset

0,2

Links

Table of n, a(n) for n=0..18.

Formula

G.f.: 1/(g^3 * (1-7*x*g^6) * (1-x*g^5)) where g = 1+x*g^7 is the g.f. of A002296.

G.f.: 1/((7-6*g) * (1-g+g^2)) where g = 1+x*g^7 is the g.f. of A002296.

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

a(n) = Sum_{k=0..n} (-1)^k * binomial(7*n+k-1,n-k).

a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(7*n-k-2,n-2*k).

This is the special case l=5, m=2, 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(5*n+2*k-3, k));

Crossrefs

Cf. A397509, A397510, A397511, A397512.

Cf. A002296, A397516.

Sequence in context: A087453 A103974 A233093 * A208588 A372215 A249930

Adjacent sequences: A397510 A397511 A397512 * A397514 A397515 A397516

Keyword

nonn,easy,new

Author

Seiichi Manyama, Jun 28 2026

Status

approved