A153978 a(n) = n*(n-1)*(n+1)*(3*n-2)/12.

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

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Jan 03 2009

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A003215, A000537, A000578, A005898, A027602, A006007, A153976, A153977, A011379, A052149, A213819.

Programs

Mathematica

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 *)

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

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)

Extensions

Edited by Bruno Berselli, Jun 15 2010

Simpler definition as suggested by Wesley Ivan Hurt, Jun 29 2014