A085540 - OEIS

%I #33 Jul 02 2023 02:18:25

%S 0,8,54,192,500,1080,2058,3584,5832,9000,13310,19008,26364,35672,

%T 47250,61440,78608,99144,123462,152000,185220,223608,267674,317952,

%U 375000,439400,511758,592704,682892,783000,893730,1015808,1149984,1297032,1457750,1632960

%N a(n) = n*(n + 1)^3.

%H Vincenzo Librandi, <a href="/A085540/b085540.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F a(n) = 2*A092364(n+1). - _Zerinvary Lajos_, May 09 2007

%F G.f.: -2*x*(4 + 7*x + x^2)/(x - 1)^5. - _R. J. Mathar_, Mar 10 2011

%F a(n) = A085537(n-1). - _Eric W. Weisstein_, Sep 08 2017

%F E.g.f.: exp(x)*x*(8 + 19*x + 9*x^2 + x^3). - _Stefano Spezia_, Jun 10 2023

%F From _Amiram Eldar_, Jul 02 2023: (Start)

%F Sum_{n>=1} 1/a(n) = 3 - Pi^2/6 - zeta(3).

%F Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/12 + 2*log(2) + 3*zeta(3)/4 - 3. (End)

%t Table[n (n + 1)^3, {n, 0, 40}] (* _Vincenzo Librandi_, Aug 15 2016 *)

%t LinearRecurrence[{5,-10,10,-5,1},{0,8,54,192,500},40] (* _Harvey P. Dale_, May 06 2019 *)

%o (Magma) [n*(n+1)^3: n in [0..40]]; // _Vincenzo Librandi_, Aug 15 2016

%Y Cf. A085537 (same sequence with a 0 prepended), A092364.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Jul 05 2003