A089670 a(n) = S3(n,2), where S3(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^3.

0, 8, 283, 6044, 101360, 1470640, 19361174, 237684384, 2768042208, 30935313600, 334481353690, 3519672963752, 36206551801264, 365363625058432, 3626585989411280, 35485636769545600, 342894590805622656, 3276865150482420480, 31008279252965786178

Offset: 0

Author

N. J. A. Sloane, Jan 04 2004

Keywords

• nonn
• easy

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), this sequence (t=2), A089671 (t=3), A089672 (t=4).

Cf. A089658, A089664.

Programs

Mathematica

a[n_]:= a[n]= Sum[k^2*(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 A089670(n): return sum(k^2*(sum(binomial(n,j) for j in (0..k)))^3 for k in (0..n))
[A089670(n) for n in (0..40)] # G. C. Greubel, May 26 2022

Formula

a(n) = Sum_{k=0..n} k^2 *(Sum_{j=0..k} binomial(n,j))^3. - G. C. Greubel, May 26 2022

a(n) ~ 7/24 * 8^n * n^3 * (1 - 9/(14*sqrt(Pi*n)) + (12/7 - 3^(3/2)/(14*Pi))/n). - Vaclav Kotesovec, May 27 2022