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%I #19 Apr 03 2021 08:43:21
%S 0,27648,720000,7776000,51861600,252887040,987614208,3265920000,
%T 9487368000,24839654400,59717623680,133689523968,281719620000,
%U 563576832000,1077621350400,1980468817920,3514388300928,6044699520000,10109900304000,16487780601600,26281368257760
%N a(n) = n * (n+1)^2 * (n+2)^3 * (n+3)^4.
%H T. D. Noe, <a href="/A101214/b101214.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).
%F G.f.: -288*x*(x^6+109*x^5+1435*x^4+4735*x^3+4780*x^2+1444*x+96) / (x-1)^11. - _Colin Barker_, Jul 04 2015
%F From _Amiram Eldar_, Apr 03 2021: (Start)
%F Sum_{n>=1} 1/a(n) = -20129/7776 + 175*Pi^2/648 + Pi^4/1080 - 5*zeta(3)/36.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 30311/7776 - 13*Pi^2/1296 - 7*Pi^4/8640 - 344*log(2)/81 - 31*zeta(3)/48. (End)
%e a(1) = 1 * (1+1)^2 * (1+2)^3 * (1+3)^4 = 27648.
%t Table[n*(n + 1)^2*(n + 2)^3*(n + 3)^4, {n, 0, 20}] (* _Stefan Steinerberger_, Feb 26 2006 *)
%o (Maxima) A101214(n):=n*(n+1)^2*(n+2)^3*(n+3)^4$ makelist(A101214(n),n,0,20); /* _Martin Ettl_, Dec 15 2012 */
%o (PARI) a(n) = n * (n+1)^2 * (n+2)^3 * (n+3)^4 \\ _Colin Barker_, Jul 04 2015
%o (PARI) concat(0, Vec(-288*x*(x^6 +109*x^5 +1435*x^4 +4735*x^3 +4780*x^2 +1444*x +96)/(x -1)^11 + O(x^100))) \\ _Colin Barker_, Jul 04 2015
%Y Cf. A101213.
%K nonn,easy
%O 0,2
%A _Parthasarathy Nambi_, Dec 13 2004
%E More terms from _Stefan Steinerberger_, Feb 26 2006
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