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%I #15 May 27 2022 07:13:15
%S 0,8,283,6044,101360,1470640,19361174,237684384,2768042208,
%T 30935313600,334481353690,3519672963752,36206551801264,
%U 365363625058432,3626585989411280,35485636769545600,342894590805622656,3276865150482420480,31008279252965786178
%N a(n) = S3(n,2), where S3(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^3.
%H G. C. Greubel, <a href="/A089670/b089670.txt">Table of n, a(n) for n = 0..1000</a>
%H 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 a(n) = Sum_{k=0..n} k^2 *(Sum_{j=0..k} binomial(n,j))^3. - _G. C. Greubel_, May 26 2022
%F 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
%t a[n_]:= a[n]= Sum[k^2*(Sum[Binomial[n, j], {j,0,k}])^3, {k,0,n}];
%t Table[a[n], {n, 0, 40}] (* _G. C. Greubel_, May 26 2022 *)
%o (SageMath)
%o def A089670(n): return sum(k^2*(sum(binomial(n,j) for j in (0..k)))^3 for k in (0..n))
%o [A089670(n) for n in (0..40)] # _G. C. Greubel_, May 26 2022
%Y Sequences of S3(n, t): A007403 (t=0), A089669 (t=1), this sequence (t=2), A089671 (t=3), A089672 (t=4).
%Y Cf. A089658, A089664.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Jan 04 2004