A382435 a(n) = Sum_{k=0..n} ( binomial(n,k) - binomial(n,k-1) )^6.

1, 1, 3, 129, 1587, 39443, 1125383, 30211457, 1107074979, 36214609683, 1433494688871, 54495716261011, 2275005440977063, 95146470595975399, 4170974287982618639, 185640304224109725569, 8492643748223480148419, 395051289603660979274339, 18726850582009755291702599

Offset: 0

Author

Seiichi Manyama, Mar 25 2025

Keywords

• nonn

Crossrefs

Cf. A131428, A382434.

Cf. A080233, A156644, A382433.

Programs

Maple

b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, add(b(x-1, y+j), j=[-1, 1])))
    end:
a:= n-> 2*add(b(n, n-2*j)^6, j=0..n/2)-1:
seq(a(n), n=0..18);  # Alois P. Heinz, Mar 25 2025

PARI

a(n) = sum(k=0, n, (binomial(n, k)-binomial(n, k-1))^6);

Python

from math import comb
def A382435(n): return (sum((comb(n,j)*(m:=n-(j<<1)+1)//(m+j))**6 for j in range((n>>1)+1))<<1)-1 # Chai Wah Wu, Mar 25 2025

Formula

a(n) = Sum_{k=0..n} A080233(n,k)^6 = Sum_{k=0..n} A156644(n,k)^6.

a(n) = 2 * A382433(n) - 1.