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 A153978 #54 Jun 10 2025 01:40:43
%S A153978 0,2,14,50,130,280,532,924,1500,2310,3410,4862,6734,9100,12040,15640,
%T A153978 19992,25194,31350,38570,46970,56672,67804,80500,94900,111150,129402,
%U A153978 149814,172550,197780,225680,256432,290224,327250,367710,411810,459762
%N A153978 a(n) = n*(n-1)*(n+1)*(3*n-2)/12.
%C A153978 Partial sums of A011379.
%C A153978 Antidiagonal sums of the convolution array A213819. - _Clark Kimberling_, Jul 04 2012
%H A153978 Vincenzo Librandi, <a href="/A153978/b153978.txt">Table of n, a(n) for n = 1..1000</a>
%H A153978 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A153978 a(n) = 2 * A001296(n-1) = (n-1)*n*(n+1)*(3*n-2)/12 (n>0). - _Bruno Berselli_, Apr 21 2010
%F A153978 a(n) = Sum_{i=1..n-1} binomial(i+1,i)*i^2. - _Enrique Pérez Herrero_, Jun 28 2014
%F A153978 G.f.: 2*x^2*(2*x+1) / (1 - x)^5. - _Colin Barker_, Jun 28 2014
%F A153978 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 4. - _Vincenzo Librandi_, Jun 30 2014
%F A153978 a(n) = Sum_{k=1..n-1}k*((n-1)*n/2 + k) for n > 1. - _J. M. Bergot_, Feb 16 2018
%F A153978 From _Amiram Eldar_, Aug 23 2022: (Start)
%F A153978 Sum_{n>=2} 1/a(n) = 141/5 - 9*sqrt(3)*Pi/5 - 81*log(3)/5.
%F A153978 Sum_{n>=2} (-1)^n/a(n) = 18*sqrt(3)*Pi/5 + 48*log(2)/5 - 129/5. (End)
%t A153978 With[{r=Range[0,50]},Accumulate[r^2+r^3]] (* _Harvey P. Dale_, Jan 16 2011 *)
%t A153978 Rest[CoefficientList[Series[-2 x^2 * (2 x + 1)/(x - 1)^5, {x, 0, 40}], x]] (* _Vincenzo Librandi_, Jun 30 2014 *)
%t A153978 LinearRecurrence[{5,-10,10,-5,1}, {0,2,14,50,130}, 25] (* _G. C. Greubel_, Sep 01 2016 *)
%o A153978 (PARI) concat(0, Vec(-2*x^2*(2*x+1)/(x-1)^5 + O(x^100))) \\ _Colin Barker_, Jun 28 2014
%o A153978 (PARI) a(n) = n*(n-1)*(n+1)*(3*n-2)/12 \\ _Charles R Greathouse IV_, Sep 01 2016
%Y A153978 Cf. A003215, A000537, A000578, A005898, A027602, A006007, A153976, A153977, A011379, A052149, A213819.
%K A153978 nonn,easy
%O A153978 1,2
%A A153978 _Vladimir Joseph Stephan Orlovsky_, Jan 03 2009
%E A153978 Edited by _Bruno Berselli_, Jun 15 2010
%E A153978 Simpler definition as suggested by _Wesley Ivan Hurt_, Jun 29 2014