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
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).
Crossrefs
Cf. A163676.
Programs
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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 *) -
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
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)
Extensions
Edited, a(12) corrected - R. J. Mathar, Aug 05 2009