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

1,2

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