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%I #21 Apr 03 2021 09:48:06
%S 0,108,1152,6000,21600,61740,150528,326592,648000,1197900,2090880,
%T 3480048,5564832,8599500,12902400,18865920,26967168,37779372,51984000,
%U 70383600,93915360,123665388,160883712,207000000,263640000,332642700
%N a(n) = n * (n+1)^2 * (n+2)^3.
%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 a(0)=0, a(1)=108, a(2)=1152, a(3)=6000, a(4)=21600, a(5)=61740, a(6)=150528, 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). - _Harvey P. Dale_, Mar 29 2012
%F G.f.: -((12*(x^4+17*x^3+33*x^2+9 x))/(x-1)^7). - _Harvey P. Dale_, Mar 29 2012
%F From _Amiram Eldar_, Apr 03 2021: (Start)
%F Sum_{n>=1} 1/a(n) = 69/16 - 3*Pi^2/8 - zeta(3)/2.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 55/16 - Pi^2/48 - 4*log(2) - 3*zeta(3)/8. (End)
%e a(1) = 1 * (1+1)^2 * (1+2)^3 = 108.
%t Table[n*(n + 1)^2*(n + 2)^3, {n, 0, 25}] (* _Stefan Steinerberger_, Feb 26 2006 *)
%t LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,108,1152,6000,21600,61740,150528},30] (* _Harvey P. Dale_, Mar 29 2012 *)
%o (Maxima) A101213(n):=n*(n+1)^2*(n+2)^3$ makelist(A101213(n),n,0,20); /* _Martin Ettl_, Dec 15 2012 */
%Y Cf. A101214.
%K nonn
%O 0,2
%A _Parthasarathy Nambi_, Dec 13 2004
%E More terms from _Stefan Steinerberger_, Feb 26 2006
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