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a(n) = n^7*(n+1)^2/2.
5

%I #14 May 14 2022 03:54:38

%S 0,2,576,17496,204800,1406250,6858432,26353376,84934656,239148450,

%T 605000000,1403076312,3027787776,6149354666,11859019200,21870000000,

%U 38788923392,66474865026,110505715776,178774347800,282240000000

%N a(n) = n^7*(n+1)^2/2.

%H G. C. Greubel, <a href="/A163277/b163277.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

%F From _R. J. Mathar_, Feb 05 2010: (Start)

%F a(n) = n^2*A163275(n).

%F G.f.: 2*x*(1 +278*x +5913*x^2 +27760*x^3 +38435*x^4 +16434*x^5 +1867*x^6 +32*x^7)/(x-1)^10. (End)

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

%F Sum_{n>=1} 1/a(n) = 16 - 7*Pi^2/3 - 4*Pi^4/45 - 4*Pi^6/945 + 10*zeta(3) + 6*zeta(5) + 2*zeta(7).

%F Sum_{n>=1} (-1)^(n+1)/a(n) = 28*log(2) + 15*zeta(3)/2 + 45*zeta(5)/8 + 63*zeta(7)/32 - 16 - 5*Pi^2/6 - 7*Pi^4/90 - 31*Pi^6/7560. (End)

%p A163277 := proc(n) n^7*(n+1)^2/2 ; end proc: seq(A163277(n),n=0..60) ; \\ _R. J. Mathar_, Feb 05 2010

%t Table[(1/2)*n^7*(n + 1)^2, {n,0,50}] (* _G. C. Greubel_, Dec 12 2016 *)

%o (PARI) concat([0], Vec(2*x*(1 +278*x +5913*x^2 +27760*x^3 +38435*x^4 +16434*x^5 +1867*x^6 +32*x^7)/(x-1)^10 + O(x^50))) \\ _G. C. Greubel_, Dec 12 2016

%o (Magma) [n^7*(n+1)^2/2: n in [0..30]]; // _Vincenzo Librandi_, Dec 13 2016

%Y Cf. A006002, A099903, A163102, A163274, A163275, A163276.

%K easy,nonn

%O 0,2

%A _Omar E. Pol_, Jul 24 2009

%E Extended by _R. J. Mathar_, Feb 05 2010