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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
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
Sequence in context: A056080 A241232 A163796 * A214908 A143553 A341493
KEYWORD
nonn,easy
AUTHOR
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.)