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 A153978 a(n) = n*(n-1)*(n+1)*(3*n-2)/12. 5
 0, 2, 14, 50, 130, 280, 532, 924, 1500, 2310, 3410, 4862, 6734, 9100, 12040, 15640, 19992, 25194, 31350, 38570, 46970, 56672, 67804, 80500, 94900, 111150, 129402, 149814, 172550, 197780, 225680, 256432, 290224, 327250, 367710, 411810, 459762 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Partial sums of A011379. Antidiagonal sums of the convolution array A213819. - Clark Kimberling, Jul 04 2012 LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..1000 Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1). FORMULA a(n) = 2 * A001296(n-1) = (n-1)*n*(n+1)*(3*n-2)/12 (n>0). - Bruno Berselli, Apr 21 2010 a(n) = Sum_{i=1..n-1} binomial(i+1,i)*i^2. - Enrique Pérez Herrero, Jun 28 2014 G.f.: 2*x^2*(2*x+1) / (1 - x)^5. - Colin Barker, Jun 28 2014 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 a(n) = Sum_{k=1..n-1}k*((n-1)*n/2 + k) for n > 1. - J. M. Bergot, Feb 16 2018 From Amiram Eldar, Aug 23 2022: (Start) Sum_{n>=2} 1/a(n) = 141/5 - 9*sqrt(3)*Pi/5 - 81*log(3)/5. Sum_{n>=2} (-1)^n/a(n) = 18*sqrt(3)*Pi/5 + 48*log(2)/5 - 129/5. (End) MATHEMATICA a[n_]:=n^2; b[n_]:=n^3; c[n_]:=b[n]+a[n]; lst={}; s=0; Do[AppendTo[lst, s+=c[n]], {n, 0, 6!}]; lst With[{r=Range[0, 50]}, Accumulate[r^2+r^3]] (* Harvey P. Dale, Jan 16 2011 *) Rest[CoefficientList[Series[-2 x^2 * (2 x + 1)/(x - 1)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 30 2014 *) LinearRecurrence[{5, -10, 10, -5, 1}, {0, 2, 14, 50, 130}, 25] (* G. C. Greubel, Sep 01 2016 *) PROG (PARI) concat(0, Vec(-2*x^2*(2*x+1)/(x-1)^5 + O(x^100))) \\ Colin Barker, Jun 28 2014 (PARI) a(n) = n*(n-1)*(n+1)*(3*n-2)/12 \\ Charles R Greathouse IV, Sep 01 2016 CROSSREFS Cf. A003215, A000537, A000578, A005898, A027602, A006007, A153976, A153977, A011379, A052149, A213819. Sequence in context: A056080 A241232 A163796 * A214908 A143553 A341493 Adjacent sequences: A153975 A153976 A153977 * A153979 A153980 A153981 KEYWORD nonn,easy AUTHOR Vladimir Joseph Stephan Orlovsky, Jan 03 2009 EXTENSIONS Edited by Bruno Berselli, Jun 15 2010 Simpler definition as suggested by Wesley Ivan Hurt, Jun 29 2014 STATUS approved

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Last modified July 20 20:50 EDT 2024. Contains 374459 sequences. (Running on oeis4.)