cp's OEIS Frontend

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

%I A089672 #11 May 27 2022 06:47:03
%S A089672 0,8,1051,47024,1343372,29595904,549599246,9039987264,135800368320,
%T A089672 1901346478080,25165027679242,318105020914208,3870088369412824,
%U A089672 45584244411107584,522235732874214800,5840992473138691072,63970901725419781632,687749464543749095424,7273214936974305201570
%N A089672 a(n) = S3(n,4), where S3(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^3.
%H A089672 G. C. Greubel, <a href="/A089672/b089672.txt">Table of n, a(n) for n = 0..1000</a>
%H A089672 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 A089672 a(n) = Sum_{k=0..n} k^4 *(Sum_{j=0..k} binomial(n,j))^3. - _G. C. Greubel_, May 26 2022
%F A089672 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
%p A089672 S3:= (n, t) -> add(k^t*add(binomial(n, j), j = 0..k)^3, k = 0..n);
%p A089672 seq(S3(n, 4), n = 0..40);
%t A089672 a[n_]:= a[n]= Sum[k^4*(Sum[Binomial[n, j], {j,0,k}])^3, {k,0,n}];
%t A089672 Table[a[n], {n, 0, 40}] (* _G. C. Greubel_, May 26 2022 *)
%o A089672 (SageMath)
%o A089672 def A089672(n): return sum(k^4*(sum(binomial(n,j) for j in (0..k)))^3 for k in (0..n))
%o A089672 [A089672(n) for n in (0..40)] # _G. C. Greubel_, May 26 2022
%Y A089672 Sequences of S3(n, t): A007403 (t=0), A089669 (t=1), A089670 (t=2), A089671 (t=3), this sequence (t=4).
%Y A089672 Cf. A089658, A089664.
%K A089672 nonn
%O A089672 0,2
%A A089672 _N. J. A. Sloane_, Jan 04 2004