A019584 - OEIS

%I #34 Feb 13 2023 02:53:24

%S 0,0,1,18,108,400,1125,2646,5488,10368,18225,30250,47916,73008,107653,

%T 154350,216000,295936,397953,526338,685900,882000,1120581,1408198,

%U 1752048,2160000,2640625,3203226,3857868,4615408,5487525,6486750,7626496,8921088,10385793

%N a(n) = n^2*(n-1)^3/4.

%H Vincenzo Librandi, <a href="/A019584/b019584.txt">Table of n, a(n) for n = 0..600</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).

%F a(n) = Sum_{j=1..n-2} Sum_{i=1..n-2} (i^3 + j^3)/2. - _Alexander Adamchuk_, Oct 24 2004

%F G.f.: x^2*(1 + 12*x + 15*x^2 + 2*x^3)/(1 - x)^6. - _Colin Barker_, May 04 2012

%F a(n) = Sum_{i=0..n-1} (n-1)*(n-1-i)^3 for n>0. - _Bruno Berselli_, Oct 31 2017

%F From _Amiram Eldar_, Feb 13 2023: (Start)

%F a(n) = A099903(n-1)/2.

%F Sum_{n>=2} 1/a(n) = 16 - 2*Pi^2 + 4*zeta(3).

%F Sum_{n>=2} (-1)^n/a(n) = 24*log(2) - 16 - Pi^2/3 + 3*zeta(3). (End)

%t Table[n^2*(n-1)^3/4, {n,0,100}]

%o (Magma) [n^2*(n-1)^3/4: n in [0..60]]; // _Vincenzo Librandi_, Apr 26 2011

%o (PARI) a(n)=n^2*(n-1)^3/4 \\ _Charles R Greathouse IV_, Oct 07 2015

%Y Cf. A099903.

%K nonn,easy

%O 0,4

%A _N. J. A. Sloane_.