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a(n) = n^4*(n+1)^2/2.
6

%I #18 May 14 2022 03:54:55

%S 0,2,72,648,3200,11250,31752,76832,165888,328050,605000,1054152,

%T 1752192,2798978,4321800,6480000,9469952,13530402,18948168,26064200,

%U 35280000,47064402,61960712,80594208,103680000,132031250,166567752,208324872

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

%C Row sums of triangle A163284.

%H Harvey P. Dale, <a href="/A163274/b163274.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).

%F From _R. J. Mathar_, Jul 29 2009: (Start)

%F a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).

%F G.f.: -2*x*(1 + 29*x + 93*x^2 + 53*x^3 + 4*x^4)/(x-1)^7. (End)

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

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

%F Sum_{n>=1} (-1)^(n+1)/a(n) = 10 + Pi^2/3 + 7*Pi^4/360 - 16*log(2) - 3*zeta(3). (End)

%t Table[(n^4 (n+1)^2)/2,{n,0,30}] (* or *) LinearRecurrence[ {7,-21,35,-35,21,-7,1},{0,2,72,648,3200,11250,31752},30] (* _Harvey P. Dale_, May 07 2012 *)

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

%Y Cf. A006002, A099903, A163102, A163275, A163276, A163277, A163284.

%K easy,nonn

%O 0,2

%A _Omar E. Pol_, Jul 24 2009

%E More terms from _R. J. Mathar_, Jul 29 2009