A382433 a(n) = S(6,n), where S(r,n) = Sum_{k=0..floor(n/2)} ( binomial(n,k) - binomial(n,k-1) )^r.

1, 1, 2, 65, 794, 19722, 562692, 15105729, 553537490, 18107304842, 716747344436, 27247858130506, 1137502720488532, 47573235297987700, 2085487143991309320, 92820152112054862785, 4246321874111740074210, 197525644801830489637170, 9363425291004877645851300

Offset: 0

Author

Seiichi Manyama, Mar 25 2025

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..562

Crossrefs

Column k=6 of A357824.

Cf. A000108, A129123.

Cf. A008315, A120730, A382435.

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-> add(b(n, n-2*j)^6, j=0..n/2):
seq(a(n), n=0..18);  # Alois P. Heinz, Mar 25 2025

Mathematica

Table[Sum[Binomial[n,k] * (Binomial[n,k] - Binomial[n,k-1])^5, {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 25 2025 *)

PARI

a(n) = sum(k=0, n, binomial(n, k)*(binomial(n, k)-binomial(n, k-1))^5);

Python

from math import comb
def A382433(n): return sum((comb(n,j)*(m:=n-(j<<1)+1)//(m+j))**6 for j in range((n>>1)+1)) # Chai Wah Wu, Mar 25 2025

Formula

a(n) = Sum_{k=0..floor(n/2)} A008315(n,k)^6.

a(n) = Sum_{k=0..n} A120730(n,k)^6.

a(n) = A357824(n,6).

a(n) = Sum_{k=0..n} binomial(n,k) * ( binomial(n,k) - binomial(n,k-1) )^5.

a(n) ~ 5 * 2^(6*n+4) / (3^(5/2) * Pi^(5/2) * n^(11/2)). - Vaclav Kotesovec, Mar 25 2025