A163276 - OEIS

%I #22 May 14 2022 03:54:43

%S 0,2,288,5832,51200,281250,1143072,3764768,10616832,26572050,60500000,

%T 127552392,252315648,473027282,847072800,1458000000,2424307712,

%U 3910286178,6139206432,9409176200,14112000000,20755401282,29988984608

%N a(n) = n^6*(n+1)^2/2.

%H G. C. Greubel, <a href="/A163276/b163276.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).

%F G.f.: 2*x*(1+135*x+1656*x^2+4456*x^3+3231*x^4+585*x^5+16*x^6)/(1-x)^9. - _Colin Barker_, May 05 2012

%F From _Amiram Eldar_, May 14 2022: (Start)

%F Sum_{n>=1} 1/a(n) = 2*Pi^2 + Pi^4/15 + 2*Pi^6/945 - 14 - 8*zeta(3) - 4*zeta(5).

%F Sum_{n>=1} (-1)^(n+1)/a(n) = 14 + 2*Pi^2/3 + 7*Pi^4/120 + 31*Pi^6/15120 - 24*log(2) - 6*zeta(3) - 15*zeta(5)/4. (End)

%p seq((1/2)*n^6*(n+1)^2, n = 0 .. 25); # _Emeric Deutsch_, Aug 01 2009

%t Table[(1/2)*n^6*(n + 1)^2, {n,0,50}] (* or *) LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1}, {0, 2, 288, 5832, 51200, 281250, 1143072, 3764768, 10616832}, 50] (* _G. C. Greubel_, Dec 12 2016 *)

%o (PARI) concat([0], Vec(2*x*(1 + 135*x +1656*x^2 +4456*x^3 +3231*x^4 +585*x^5 +16*x^6)/(1-x)^9 + O(x^50))) \\ _G. C. Greubel_, Dec 12 2016

%o (Magma) [n^6*(n+1)^2/2: n in [0..30]]; // _Vincenzo Librandi_, Dec 13 2016

%Y Cf. A006002, A099903, A163102, A163274, A163275, A163277.

%K easy,nonn

%O 0,2

%A _Omar E. Pol_, Jul 24 2009

%E Extended by _Emeric Deutsch_, Aug 01 2009