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a(n) = S1(n,2), where S1(n, t) = Sum_{k=0..n} (k^t * Sum_{j=0..k} binomial(n,j)).
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%I #23 May 25 2022 02:08:17

%S 0,2,19,104,440,1600,5264,16128,46848,130560,352000,923648,2369536,

%T 5963776,14766080,36044800,86900736,207224832,489357312,1145569280,

%U 2660761600,6136266752,14060355584,32027705344,72561459200,163577856000,367068708864,820204535808

%N a(n) = S1(n,2), where S1(n, t) = Sum_{k=0..n} (k^t * Sum_{j=0..k} binomial(n,j)).

%H Vincenzo Librandi, <a href="/A089659/b089659.txt">Table of n, a(n) for n = 0..1000</a>

%H 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 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-24,32,-16).

%F a(n) = 2^(n-3)*n*(7*n^2 + 12*n + 5)/3. (see Wang and Zhang p. 333)

%F From _Chai Wah Wu_, Jun 21 2016: (Start)

%F a(n) = 8*a(n-1) - 24*a(n-2) + 32*a(n-3) - 16*a(n-4) for n > 3.

%F G.f.: x*(2 + 3*x)/(1 - 2*x)^4. (End)

%F E.g.f.: x*(12 + 33*x + 14*x^2)*exp(2*x)/6. - _Ilya Gutkovskiy_, Jun 21 2016

%t LinearRecurrence[{8,-24,32,-16}, {0,2,19,104}, 40] (* _Vincenzo Librandi_, Jun 22 2016 *)

%o (Magma) I:=[0,2,19,104]; [n le 4 select I[n] else 8*Self(n-1)-24*Self(n-2)+32*Self(n-3)-16*Self(n-4): n in [1..41]]; // _Vincenzo Librandi_, Jun 22 2016

%o (SageMath) [2^(n-3)*n*(7*n^2 + 12*n + 5)/3 for n in (0..40)] # _G. C. Greubel_, May 24 2022

%Y Columns : A000012, A000012, A000984, A006480, A008977, A008978.

%Y Sequences of S1(n, t): A001792 (t=0), A089658 (t=1), this sequence (t=2), A089660 (t=3), A089661 (t=4), A089662 (t=5), A089663 (t=6).

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Jan 04 2004