A163683 a(n) = n^2*(2*n + 5).

0, 7, 36, 99, 208, 375, 612, 931, 1344, 1863, 2500, 3267, 4176, 5239, 6468, 7875, 9472, 11271, 13284, 15523, 18000, 20727, 23716, 26979, 30528, 34375, 38532, 43011, 47824, 52983, 58500, 64387, 70656, 77319, 84388, 91875, 99792, 108151, 116964, 126243, 136000

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

Row sums from A163676: a(n) = Sum_{m=1..n} (4*m*n + 2*m + 2*n - 1).

G.f.: x*(7 + 8*x - 3*x^2)/(1 - x)^4.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).

E.g.f.: x*(7 + 11*x + 2*x^2)*exp(x). - G. C. Greubel, Aug 02 2017

From Amiram Eldar, Feb 25 2023: (Start)

Sum_{n>=1} 1/a(n) = Pi^2/30 + 4*log(2)/25 - 92/375.

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/60 - Pi/25 -2*log(2)/25 + 52/375. (End)

Mathematica

CoefficientList[Series[-x*(-7-8*x+3*x^2)/(x-1)^4, {x, 0, 40}], x] (* or *) LinearRecurrence[{4, -6, 4, -1}, {0, 7, 36, 99}, 50](* Vincenzo Librandi, Mar 06 2012 *)
Table[n^2(2n+5), {n, 0, 50}] (* Harvey P. Dale, Apr 10 2019 *)

Prog

(PARI) my(x='x+O('x^50)); concat([0], Vec(x*(7 +8*x -3*x^2)/(1 - x)^4)) \\ G. C. Greubel, Aug 02 2017

Crossrefs

Cf. A163676.

Sequence in context: A176543 A018199 A063168 * A103090 A250414 A212141

Adjacent sequences: A163680 A163681 A163682 * A163684 A163685 A163686

Keyword

nonn,easy

Author

Vincenzo Librandi, Aug 03 2009

Extensions

Edited, a(12) corrected - R. J. Mathar, Aug 05 2009

Status

approved