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A089658 a(n) = S1(n,1), where S1(n, t) = Sum_{k=0..n} (k^t * Sum_{j=0..k} binomial(n,j)).

%I A089658 #24 May 25 2022 02:08:43
%S A089658 0,2,11,42,136,400,1104,2912,7424,18432,44800,107008,251904,585728,
%T A089658 1347584,3072000,6946816,15597568,34799616,77201408,170393600,
%U A089658 374341632,818937856,1784676352,3875536896,8388608000,18102616064,38956695552,83617644544,179046449152
%N A089658 a(n) = S1(n,1), where S1(n, t) = Sum_{k=0..n} (k^t * Sum_{j=0..k} binomial(n,j)).
%H A089658 Vincenzo Librandi, <a href="/A089658/b089658.txt">Table of n, a(n) for n = 0..1000</a>
%H A089658 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 A089658 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-12,8).
%F A089658 a(n) = n*(5 + 3*n) * 2^(n-3). (See Wang and Zhang p. 333.)
%F A089658 From _Chai Wah Wu_, Jun 21 2016: (Start)
%F A089658 a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3) for n > 2.
%F A089658 G.f.: x*(2 - x)/(1 - 2*x)^3. (End)
%F A089658 E.g.f.: x*(4 + 3*x)*exp(2*x)/2. - _Ilya Gutkovskiy_, Jun 21 2016
%F A089658 a(n) = 2*A001788(n) - A001788(n-1). - _R. J. Mathar_, Jul 22 2021
%t A089658 LinearRecurrence[{6,-12,8}, {0,2,11}, 40] (* _Vincenzo Librandi_, Jun 22 2016 *)
%o A089658 (Magma) I:=[0,2,11]; [n le 3 select I[n] else 6*Self(n-1)-12*Self(n-2)+8*Self(n-3): n in [1..41]]; // _Vincenzo Librandi_, Jun 22 2016
%o A089658 (SageMath) [n*(5+3*n)*2^(n-3) for n in (0..40)] # _G. C. Greubel_, May 24 2022
%Y A089658 Sequences of S1(n, t): A001792 (t=0), this sequence (t=1), A089659 (t=2), A089660 (t=3), A089661 (t=4), A089662 (t=5), A089663 (t=6).
%K A089658 nonn,easy
%O A089658 0,2
%A A089658 _N. J. A. Sloane_, Jan 04 2004