A089671 a(n) = S3(n,3), where S3(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^3.

0, 8, 539, 16632, 364042, 6510160, 101817234, 1447146512, 19144522160, 239513659776, 2865559784050, 33052451375152, 369790434398988, 4031805422883680, 42996629236138928, 449821323139340160, 4627609615665499456, 46907404618252667392, 469254490696475078130

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

Cf. A089658, A089664.

Programs

Mathematica

a[n_]:= a[n]= Sum[k^3*(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 A089671(n): return sum(k^3*(sum(binomial(n,j) for j in (0..k)))^3 for k in (0..n))
[A089671(n) for n in (0..40)] # G. C. Greubel, May 26 2022

Formula

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

a(n) ~ 15 * 2^(3*n-6) * n^4 * (1 - 2/(5*sqrt(Pi*n)) + (2 - sqrt(3)/(5*Pi))/n). - Vaclav Kotesovec, May 27 2022