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A089660 a(n) = S1(n,3), where S1(n, t) = Sum_{k=0..n} (k^t * Sum_{j=0..k} binomial(n,j)).

%I A089660 #22 May 25 2022 02:07:47
%S A089660 0,2,35,276,1522,6820,26664,94640,312512,975744,2913280,8386048,
%T A089660 23416320,63724544,169637888,443043840,1137934336,2879979520,
%U A089660 7194083328,17761304576,43390730240,104997322752,251881062400,599482433536,1416470986752,3324615065600
%N A089660 a(n) = S1(n,3), where S1(n, t) = Sum_{k=0..n} (k^t * Sum_{j=0..k} binomial(n,j)).
%H A089660 Vincenzo Librandi, <a href="/A089660/b089660.txt">Table of n, a(n) for n = 0..1000</a>
%H A089660 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 A089660 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (10,-40,80,-80,32).
%F A089660 a(n) = n*(15*n^3 + 30*n^2 + 21*n - 2)*2^(n-6). - _R. J. Mathar_, Sep 16 2009
%F A089660 G.f.: x*(2 + 15*x + 6*x^2 + 2*x^3)/(1-2*x)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Jul 28 2009
%F A089660 a(n) = 10*a(n-1) - 40*a(n-2) + 80*a(n-3) - 80*a(n-4) + 32*a(n-5) for n > 4. - _Chai Wah Wu_, Jun 21 2016
%F A089660 E.g.f.: x*(8 + 54*x + 60*x^2 + 15*x^3)*exp(2*x)/4. - _Ilya Gutkovskiy_, Jun 21 2016
%t A089660 LinearRecurrence[{10,-40,80,-80,32}, {0,2,35,276,1522}, 40] (* _Vincenzo Librandi_, Jun 22 2016 *)
%o A089660 (Magma) [2^(n-6)*n*(3*n*(7+10*n+5*n^2) -2): n in [0..40]]; // _G. C. Greubel_, May 24 2022
%o A089660 (SageMath) [n*(15*n^3+30*n^2+21*n-2)*2^(n-6) for n in (0..40)] # _G. C. Greubel_, May 24 2022
%Y A089660 Sequences of S1(n, t): A001792 (t=0), A089658 (t=1), A089659 (t=2), this sequence (t=3), A089661 (t=4), A089662 (t=5), A089663 (t=6).
%K A089660 nonn,easy
%O A089660 0,2
%A A089660 _N. J. A. Sloane_, Jan 04 2004