A294435 a(n) = Sum_{m=0..n} (Sum_{k=0..m} binomial(n,k))^4.

1, 17, 338, 6754, 131428, 2495906, 46434532, 849488292, 15328171208, 273445276258, 4831735919236, 84688295720132, 1474133269832776, 25506505928857892, 439034457665156168, 7522356118216054216, 128364598453699389840, 2182553210810903666402, 36989251585608710893636

Offset: 0

Author

N. J. A. Sloane, Nov 17 2017

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..300
• N. J. Calkin, A curious binomial identity, Discr. Math., 131 (1994), 335-337.
• M. Hirschhorn, Calkin's binomial identity, Discr. Math., 159 (1996), 273-278.

Crossrefs

Same expression with exponent b instead of 4: A001792 (b=1), A003583 (b=2), A007403 (b=3), A294435 (b=4), A294436 (b=5).

Programs

Maple

A:=proc(n,k) local j; add(binomial(n,j),j=0..k); end;
S:=proc(n,p) local i; global A; add(A(n,i)^p, i=0..n); end;
[seq(S(n,4),n=0..30)];

Mathematica

Table[Sum[Sum[Binomial[n,k], {k,0,m}]^4, {m,0,n}], {n,0,15}] (* Vaclav Kotesovec, Jun 07 2019 *)

PARI

a(n) = sum(m=0, n, sum(k=0, m, binomial(n,k))^4); \\ Michel Marcus, Nov 18 2017

Formula

a(n) ~ n * 2^(4*n - 1). - Vaclav Kotesovec, Jun 07 2019