A089672 a(n) = S3(n,4), where S3(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^3.

0, 8, 1051, 47024, 1343372, 29595904, 549599246, 9039987264, 135800368320, 1901346478080, 25165027679242, 318105020914208, 3870088369412824, 45584244411107584, 522235732874214800, 5840992473138691072, 63970901725419781632, 687749464543749095424, 7273214936974305201570

Offset: 0

Author

N. J. A. Sloane, Jan 04 2004

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Jun Wang and Zhizheng Zhang, On extensions of Calkin's binomial identities, Discrete Math., 274 (2004), 331-342.

Crossrefs

Sequences of S3(n, t): A007403 (t=0), A089669 (t=1), A089670 (t=2), A089671 (t=3), this sequence (t=4).

Cf. A089658, A089664.

Programs

Maple

S3:= (n, t) -> add(k^t*add(binomial(n, j), j = 0..k)^3, k = 0..n);
seq(S3(n, 4), n = 0..40);

Mathematica

a[n_]:= a[n]= Sum[k^4*(Sum[Binomial[n, j], {j,0,k}])^3, {k,0,n}];
Table[a[n], {n, 0, 40}] (* G. C. Greubel, May 26 2022 *)

SageMath

def A089672(n): return sum(k^4*(sum(binomial(n,j) for j in (0..k)))^3 for k in (0..n))
[A089672(n) for n in (0..40)] # G. C. Greubel, May 26 2022

Formula

a(n) = Sum_{k=0..n} k^4 *(Sum_{j=0..k} binomial(n,j))^3. - G. C. Greubel, May 26 2022

a(n) ~ 31 * 2^(3*n - 5) * n^5 / 5 * (1 - 15/(62*sqrt(Pi*n)) + (75 - 5*sqrt(3)/Pi) / (31*n)). - Vaclav Kotesovec, May 27 2022