A397567 a(n) = Sum_{k=0..n} binomial(4*n+2*k-2,k).

1, 5, 54, 664, 8625, 115506, 1577311, 21833033, 305238726, 4300158499, 60947481440, 868075247727, 12414285158328, 178143045874721, 2563779890825235, 36989976295984049, 534858504103606897, 7748732574979636374, 112451247125555173765, 1634406098930351324291

Offset

0,2

Links

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

Formula

G.f.: 1/(g^2 * (1-6*x*g^5) * (1-x*g^4)) where g = 1+x*g^6 is the g.f. of A002295.

G.f.: g/((6-5*g) * (1-g+g^2)) where g = 1+x*g^6 is the g.f. of A002295.

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

a(n) = Sum_{k=0..n} (-1)^k * binomial(6*n+k,n-k).

a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(6*n-k-1,n-2*k).

This is the special case l=4, m=2, c=1, 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(4*n+2*k-2, k));

Crossrefs

Cf. A397512, A396691, A397568.

Cf. A002295.

Sequence in context: A110430 A110432 A215542 * A304322 A132619 A251583

Adjacent sequences: A397563 A397564 A397565 * A397568 A397575 A397576

Keyword

nonn,easy,new

Author

Seiichi Manyama, Jul 01 2026

Status

approved