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A089662 a(n) = S1(n,5), where S1(n,t) = Sum_{k=0..n} k^t * Sum_{j=0..k} binomial(n,j).

%I A089662 #18 May 24 2022 18:27:07
%S A089662 0,2,131,2172,20386,138580,763824,3631712,15470144,60527232,221297920,
%T A089662 765580288,2529498624,8039103488,24713744384,73818562560,215011065856,
%U A089662 612515381248,1710842904576,4695105732608,12682107944960,33768108982272,88748191645696
%N A089662 a(n) = S1(n,5), where S1(n,t) = Sum_{k=0..n} k^t * Sum_{j=0..k} binomial(n,j).
%H A089662 Vincenzo Librandi, <a href="/A089662/b089662.txt">Table of n, a(n) for n = 0..1000</a>
%H A089662 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.
%H A089662 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (14,-84,280,-560,672,-448,128).
%F A089662 There is an explicit formula for the sum - see Wang and Zhang, p. 334.
%F A089662 From _Chai Wah Wu_, Jun 21 2016: (Start)
%F A089662 a(n) = 14*a(n-1) - 84*a(n-2) + 280*a(n-3) - 560*a(n-4) + 672*a(n-5) - 448*a(n-6) + 128*a(n-7) for n > 6.
%F A089662 G.f.: x*(-16*x^5 + 64*x^4 + 422*x^3 + 506*x^2 + 103*x + 2)/(1 - 2*x)^7. (End)
%F A089662 a(n) = 2^(n-7)*n*(21*n^5 + 61*n^4 + 55*n^3 + 15*n^2 - 28*n + 4). - _Ilya Gutkovskiy_, Jun 21 2016
%F A089662 E.g.f.: (1/4)*x*(8 + 246*x + 940*x^2 + 1015*x^3 + 376*x^4 + 42*x^5)*exp(2*x). - _G. C. Greubel_, May 24 2022
%t A089662 LinearRecurrence[{14,-84,280,-560,672,-448,128}, {0,2,131,2172,20386,138580, 763824}, 30] (* _Vincenzo Librandi_, Jun 22 2016 *)
%o A089662 (Magma) [2^(n-7)*n*(21*n^5+61*n^4+55*n^3+15*n^2-28*n+4): n in [0..30]]; // _Vincenzo Librandi_, Jun 22 2016
%o A089662 (SageMath) [2^(n-7)*n*(21*n^5 +61*n^4 +55*n^3 +15*n^2 -28*n +4) for n in (0..40)] # _G. C. Greubel_, May 24 2022
%Y A089662 Sequences of S1(n,t): A001792 (t=0), A089658 (t=1), A089659 (t=2), A089660 (t=3), A089661 (t=4), this sequence (t=5), A089663 (t=6).
%K A089662 nonn
%O A089662 0,2
%A A089662 _N. J. A. Sloane_, Jan 04 2004