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

%I A089671 #10 May 27 2022 07:25:00
%S A089671 0,8,539,16632,364042,6510160,101817234,1447146512,19144522160,
%T A089671 239513659776,2865559784050,33052451375152,369790434398988,
%U A089671 4031805422883680,42996629236138928,449821323139340160,4627609615665499456,46907404618252667392,469254490696475078130
%N A089671 a(n) = S3(n,3), where S3(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^3.
%H A089671 G. C. Greubel, <a href="/A089671/b089671.txt">Table of n, a(n) for n = 0..1000</a>
%H A089671 Jun Wang and Zhizheng Zhang, <a href="http://dx.doi.org/10.1016/S0012-365X(03)00206-1">On extensions of Calkin's binomial identities</a>, Discrete Math., 274 (2004), 331-342.
%F A089671 a(n) = Sum_{k=0..n} k^3 *(Sum_{j=0..k} binomial(n,j))^3. - _G. C. Greubel_, May 26 2022
%F A089671 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
%t A089671 a[n_]:= a[n]= Sum[k^3*(Sum[Binomial[n, j], {j,0,k}])^3, {k,0,n}];
%t A089671 Table[a[n], {n, 0, 40}] (* _G. C. Greubel_, May 26 2022 *)
%o A089671 (SageMath)
%o A089671 def A089671(n): return sum(k^3*(sum(binomial(n,j) for j in (0..k)))^3 for k in (0..n))
%o A089671 [A089671(n) for n in (0..40)] # _G. C. Greubel_, May 26 2022
%Y A089671 Sequences of S3(n, t): A007403 (t=0), A089669 (t=1), A089670 (t=2), this sequence (t=3), A089672 (t=4).
%Y A089671 Cf. A089658, A089664.
%K A089671 nonn
%O A089671 0,2
%A A089671 _N. J. A. Sloane_, Jan 04 2004