A089661 - OEIS

%I #22 May 25 2022 02:07:12

%S 0,2,67,764,5492,30304,140672,577920,2167680,7577088,25037056,

%T 79016960,240028672,705961984,2019713024,5641535488,15431565312,

%U 41438281728,109462880256,284942925824,732004876288,1858158460928,4665915736064,11600782163968,28582042664960

%N a(n) = S1(n,4), where S1(n,t) = Sum_{k=0..n} k^t * Sum_{j=0..k} binomial(n,j).

%H Vincenzo Librandi, <a href="/A089661/b089661.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_06">Index entries for linear recurrences with constant coefficients</a>, signature (12,-60,160,-240,192,-64).

%F a(n) = (1/15)*n*(n+1)*(93*n^3 + 132*n^2 + 53*n - 38)*2^(n-5). (See Wang and Zhang, p. 334)

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

%F a(n) = 12*a(n-1) - 60*a(n-2) + 160*a(n-3) - 240*a(n-4) + 192*a(n-5) - 64*a(n-6) for n > 5.

%F G.f.: x*(2 + 43*x + 80*x^2 + 24*x^3)/(1 - 2*x)^6. (End)

%F a(n) = 2^(n-5)*n*(93*n^4 + 225*n^3 + 185*n^2 + 15*n - 38)/15. - _Ilya Gutkovskiy_, Jun 21 2016

%F E.g.f.: (1/30)*x*(60 + 885*x + 1930*x^2 + 1155*x^3 + 186*x^4)*exp(2*x). - _G. C. Greubel_, May 24 2022

%t LinearRecurrence[{12,-60,160,-240,192,-64}, {0,2,67,764,5492,30304}, 40] (* _Vincenzo Librandi_, Jun 22 2016 *)

%o (Magma) [2^(n-5)*n*(93*n^4+225*n^3+185*n^2+15*n-38)/15: n in [0..30]]; // _Vincenzo Librandi_, Jun 22 2016

%o (SageMath) [n*(n+1)*(93*n^3 +132*n^2 +53*n -38)*2^(n-5)/15 for n in (0..40)] # _G. C. Greubel_, May 24 2022

%Y Sequences of S1(n,t): A001792 (t=0), A089658 (t=1), A089659 (t=2), A089660 (t=3), this sequence (t=4), A089662 (t=5), A089663 (t=6).

%K nonn,easy

%O 0,2

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