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 A059338 #25 Jul 02 2025 16:02:00
%S A059338 1,34,342,2192,11000,47232,181888,646144,2156544,6848000,20877824,
%T A059338 61526016,176171008,492126208,1345536000,3610247168,9526771712,
%U A059338 24769069056,63546720256,161087488000,403925630976,1002841309184,2467290939392,6019866427392,14575206400000,35039249170432
%N A059338 a(n) = Sum_{k=1..n} k^5 * binomial(n,k).
%D A059338 Finding a closed form for the sum was Problem 541 in the Fall 2000 issue of The Pentagon (official journal of Kappa Mu Epsilon).
%H A059338 Harry J. Smith, <a href="/A059338/b059338.txt">Table of n, a(n) for n = 1..200</a>
%H A059338 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (12,-60,160,-240,192,-64).
%F A059338 The closed form comes from starting with (1+x)^n and repeatedly differentiating and multiplying by x. After five differentiations, substitute x=1 and get a(n) = 2^(n-5)*n^2*(n^3+10n^2+15n-10).
%F A059338 G.f.: x*(16*x^4-32*x^3-6*x^2+22*x+1)/(2*x-1)^6. - _Colin Barker_, Sep 20 2012
%p A059338 with(combinat): for n from 1 to 70 do printf(`%d,`,sum(k^5*binomial(n,k), k=1..n)) od:
%t A059338 Table[Sum[k^5*Binomial[n, k], {k,1,n}], {n,1,5}] (* or *) LinearRecurrence[{12, -60, 160, -240, 192, -64}, {1, 34, 342, 2192,
%t A059338 11000, 47232}, 10] (* _G. C. Greubel_, Jan 07 2017 *)
%o A059338 (PARI) a(n) = { sum(k=1, n, k^5*binomial(n, k)) } \\ _Harry J. Smith_, Jun 26 2009
%o A059338 (PARI) Vec(x*(16*x^4-32*x^3-6*x^2+22*x+1)/(2*x-1)^6 + O(x^25)) \\ _G. C. Greubel_, Jan 07 2017
%Y A059338 Binomial transform of A000584.
%K A059338 nonn,easy
%O A059338 1,2
%A A059338 Pat Costello (matcostello(AT)acs.eku.edu), Jan 26 2001
%E A059338 More terms from _James Sellers_, Jan 29 2001