A294436 a(n) = Sum_{m=0..n} (Sum_{k=0..m} binomial(n,k))^5.

1, 33, 1268, 50600, 1972128, 75121312, 2803732096, 102885494016, 3722920064000, 133152625650176, 4715897847097344, 165643005814853632, 5776871664703455232, 200235592430802124800, 6903358709034568712192, 236882142098621090889728, 8094539021386254685569024

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 5: A001792 (b=1), A003583 (b=2), A007403 (b=3), A294435 (b=4).

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,5),n=0..30)];

Mathematica

Table[Sum[Sum[Binomial[n,k], {k,0,m}]^5, {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))^5); \\ Michel Marcus, Nov 18 2017

Formula

a(n) ~ n * 2^(5*n - 1). - Vaclav Kotesovec, Jun 07 2019