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A089669 a(n) = S3(n,1), where S3(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^3.

%I A089669 #13 May 27 2022 06:22:40
%S A089669 0,8,155,2286,29296,344140,3807774,40327280,413058080,4120742808,
%T A089669 40242188170,386141947972,3650905945872,34087726136672,
%U A089669 314844824466704,2880757518523200,26141327872575616,235490128979282224,2107598857648209954,18752794473550896332
%N A089669 a(n) = S3(n,1), where S3(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^3.
%H A089669 G. C. Greubel, <a href="/A089669/b089669.txt">Table of n, a(n) for n = 0..1000</a>
%H A089669 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 A089669 a(n) = Sum_{k=0..n} k * (Sum_{j=0..k} binomial(n,j))^3. - _G. C. Greubel_, May 26 2022
%F A089669 From _Vaclav Kotesovec_, May 27 2022: (Start)
%F A089669 Recurrence: (n-3)*(n-2)*(n-1)*(81*n^5 - 1080*n^4 + 5769*n^3 - 15146*n^2 + 19080*n - 9088)*a(n) = (n-3)*(1863*n^7 - 29457*n^6 + 195435*n^5 - 696271*n^4 + 1410606*n^3 - 1569664*n^2 + 815328*n - 103936)*a(n-1) - 12*(1134*n^8 - 20709*n^7 + 162279*n^6 - 708529*n^5 + 1865571*n^4 - 2976218*n^3 + 2709336*n^2 - 1189824*n + 153600)*a(n-2) + 64*(405*n^8 - 7101*n^7 + 53712*n^6 - 228226*n^5 + 590469*n^4 - 934993*n^3 + 856278*n^2 - 392944*n + 65280)*a(n-3) + 256*(n-4)*(n-2)*(2*n - 7)*(81*n^5 - 675*n^4 + 2259*n^3 - 3509*n^2 + 2180*n - 384)*a(n-4).
%F A089669 a(n) ~ 3 * n^2 * 8^(n-1) * (1 - 1/sqrt(Pi*n) + (5/3 - 1/(2*Pi*sqrt(3)))/n). (End)
%t A089669 a[n_]:= a[n]= Sum[k*(Sum[Binomial[n, j], {j,0,k}])^3, {k,0,n}];
%t A089669 Table[a[n], {n,0,40}] (* _G. C. Greubel_, May 26 2022 *)
%o A089669 (SageMath)
%o A089669 @CachedFunction
%o A089669 def A089669(n): return sum(k*(sum(binomial(n,j) for j in (0..k)))^3 for k in (0..n))
%o A089669 [A089669(n) for n in (0..40)] # _G. C. Greubel_, May 26 2022
%Y A089669 Sequences of S3(n, t): A007403 (t=0), this sequence (t=1), A089670 (t=2), A089671 (t=3), A089672 (t=4).
%Y A089669 Cf. A089658, A089664.
%K A089669 nonn,easy
%O A089669 0,2
%A A089669 _N. J. A. Sloane_, Jan 04 2004