A119771 - OEIS

%I #15 Feb 13 2023 02:55:33

%S 0,1,16,90,320,875,2016,4116,7680,13365,22000,34606,52416,76895,

%T 109760,153000,208896,280041,369360,480130,616000,781011,979616,

%U 1216700,1497600,1828125,2214576,2663766,3183040,3780295,4464000,5243216,6127616,7127505,8253840

%N Product of n^2 and n-th tetrahedral number: a(n) = n^3*(n+1)*(n+2)/6.

%C If n is divisible by 10, then a(n) is divisible by 1000.

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

%F From _Alois P. Heinz_, Feb 10 2023: (Start)

%F a(n) = Sum_{k=0..n} k^2 * A061579(n,k).

%F G.f.: x*(x+1)*(9*x+1)/(x-1)^6. (End)

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

%F Sum_{n>=1} 1/a(n) = 39/8 - 3*Pi^2/4 + 3*zeta(3).

%F Sum_{n>=1} (-1)^(n+1)/a(n) = 12*log(2) - 51/8 - 3*Pi^2/8 + 9*zeta(3)/4. (End)

%e a(25) = n^3*(n+1)*(n+2)/6 = 25^3*(25+1)*(25+2)/6 = 15625*26*27/6 = 15625*13*9 = 1828125.

%p with(combinat):a:=n->sum(sum(sum(binomial(n+2,2)/3, j=1..n), k=1..n),m=1..n): seq(a(n), n=0..31); # _Zerinvary Lajos_, May 30 2007

%t a[n_] := n^3*(n+1)*(n+2)/6; Array[a, 35, 0] (* _Amiram Eldar_, Feb 13 2023 *)

%Y Cf. A000290, A000292, A061579.

%K easy,nonn

%O 0,3

%A Brandon Ang (xyz1236(AT)verizon.net), Jun 28 2006