A163683 - OEIS

%I #25 Feb 25 2023 06:32:05

%S 0,7,36,99,208,375,612,931,1344,1863,2500,3267,4176,5239,6468,7875,

%T 9472,11271,13284,15523,18000,20727,23716,26979,30528,34375,38532,

%U 43011,47824,52983,58500,64387,70656,77319,84388,91875,99792,108151,116964,126243,136000

%N a(n) = n^2*(2*n + 5).

%H Vincenzo Librandi, <a href="/A163683/b163683.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

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

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

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

%F E.g.f.: x*(7 + 11*x + 2*x^2)*exp(x). - _G. C. Greubel_, Aug 02 2017

%F From _Amiram Eldar_, Feb 25 2023: (Start)

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

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

%t 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 *)

%t Table[n^2(2n+5),{n,0,50}] (* _Harvey P. Dale_, Apr 10 2019 *)

%o (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

%Y Cf. A163676.

%K nonn,easy

%O 0,2

%A _Vincenzo Librandi_, Aug 03 2009

%E Edited, a(12) corrected - _R. J. Mathar_, Aug 05 2009