A397668 a(n) = Sum_{k=0..n} binomial(5*k-6,k).

1, 0, 6, 90, 1091, 12719, 147315, 1708095, 19864299, 231779431, 2713036209, 31848952473, 374855841243, 4422232192863, 52277932151679, 619155619085215, 7345193044897651, 87268389808287631, 1038242650597854421, 12367296045642507601, 147481415192601895036

Offset

0,3

Links

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

Formula

G.f.: 1/(g^6 * (1-5*x*g^4) * (1-x)) where g = 1+x*g^5 is the g.f. of A002294.

G.f.: 1/((5-4*g) * (1-g+g^5)) where g = 1+x*g^5 is the g.f. of A002294.

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

This is the special case l=0, 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.

4*n*(4*n-6)*(4*n-7)*(4*n-9)*a(n) - (n-1)*(3*n-5)*(23*n-42)*(49*n-72)*a(n-1) + 5*(5*n-6)*(5*n-7)*(5*n-8)*(5*n-9)*a(n-2) = 0 for n > 2.

Prog

(PARI) a(n) = sum(k=0, n, binomial(5*k-6, k));

Crossrefs

Partial sums of A397671.

Cf. A371965, A397666, A397667.

Cf. A002294, A225615, A397537.

Sequence in context: A336042 A353230 A317487 * A037959 A247150 A201073

Adjacent sequences: A397663 A397666 A397667 * A397669 A397670 A397671

Keyword

nonn,easy,new

Author

Seiichi Manyama, Jul 04 2026

Status

approved