This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A089663 #23 May 25 2022 02:06:33
%S A089663 0,2,259,6284,77180,646960,4235864,23313408,112793088,493969920,
%T A089663 1998346240,7577934848,27232132096,93517705216,308908943360,
%U A089663 986642513920,3059995508736,9247515082752,27310549696512,79012328898560,224396746424320,626707269681152
%N A089663 a(n) = S1(n, 6), where S1(n, t) = Sum_{k=0..n} (k^t * Sum_{j=0..k} binomial(n,j)).
%H A089663 G. C. Greubel, <a href="/A089663/b089663.txt">Table of n, a(n) for n = 0..1000</a>
%H A089663 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 A089663 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (16,-112,448,-1120,1792,-1792,1024,-256).
%F A089663 a(n) = (1/21)*n*(n+1)*(381*n^5 + 921*n^4 + 381*n^3 + 39*n^2 - 746*n + 368) * 2^(n-7). (See Wang and Zhang, p. 334.)
%F A089663 From _Chai Wah Wu_, Jun 21 2016: (Start)
%F A089663 a(n) = 16*a(n-1) - 112*a(n-2) + 448*a(n-3) - 1120*a(n-4) + 1792*a(n-5) - 1792*a(n-6) + 1024*a(n-7) - 256*a(n-8) for n > 7.
%F A089663 G.f.: x*(2 + 227*x + 2364*x^2 + 4748*x^3 + 2096*x^4 - 72*x^5)/(1 - 2*x)^8. (End)
%F A089663 a(n) = 2^(n-7)*n*(381*n^6 + 1302*n^5 + 1302*n^4 + 420*n^3 - 707*n^2 - 378*n + 368)/21. - _Ilya Gutkovskiy_, Jun 21 2016
%F A089663 E.g.f.: (1/42)*x*(84 + 5271*x + 33278*x^2 + 57855*x^3 + 37086*x^4 + 9303*x^5 +
%F A089663 762*x^6)*exp(2*x). - _G. C. Greubel_, May 24 2022
%t A089663 LinearRecurrence[{16,-112,448,-1120,1792,-1792,1024,-256}, {0, 2, 259, 6284, 77180, 646960, 4235864, 23313408}, 40] (* _G. C. Greubel_, Jun 22 2016 *)
%o A089663 (Magma) [2^(n-7)*n*(381*n^6+1302*n^5+1302*n^4+420*n^3-707*n^2-378*n+368)/21: n in [0..40]]; // _G. C. Greubel_, May 24 2022
%o A089663 (SageMath) [n*(n+1)*(381*n^5 +921*n^4 +381*n^3 +39*n^2 -746*n +368)*2^(n-7)/21 for n in (0..40)] # _G. C. Greubel_, May 24 2022
%Y A089663 Sequences of S1(n, t): A001792 (t=0), A089658 (t=1), A089659 (t=2), A089660 (t=3), A089661 (t=4), A089662 (t=5), this sequence (t=6).
%Y A089663 Cf. A089664, A089669.
%K A089663 nonn,easy
%O A089663 0,2
%A A089663 _N. J. A. Sloane_, Jan 04 2004