A348457 a(n) = Sum_{s=0..n} (-1)^s * ( Sum_{k=0..s} binomial(n,k) )^3.

1, -7, 38, -232, 1928, -16672, 133508, -1044736, 8337920, -67162624, 537953708, -4294193152, 34336008272, -274889261056, 2199555817952, -17592017354752, 140725278645248, -1125902437187584, 9007484455265852, -72057555801604096, 576453982622834768, -4611686599619510272, 36893651043755447672, -295147896302964047872

Offset: 0

Author

N. J. A. Sloane, Oct 28 2021

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000
• Z. Zhang, A binomial identity related to Calkin's, Discr. Math., 196 (1999), 287-289.

Crossrefs

Cf. A007403.

Programs

Maple

f := n -> add( (-1)^s*( add(binomial(n,k),k=0..s)^3 ),s=0..n);
[seq(f(n),n=0..50)];

Mathematica

a[n_] := Sum[(-1)^m * Sum[Binomial[n, k], {k, 0, m}]^3, {m, 0, n}]; Array[a, 24, 0] (* Amiram Eldar, Oct 28 2021 *)

PARI

a(n) = sum(s=0, n, (-1)^s*sum(k=0, s, binomial(n,k))^3); \\ Michel Marcus, Oct 28 2021

Formula

a(n) ~ (-1)^n * 2^(3*n-1). - Vaclav Kotesovec, Nov 01 2021