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a(n) = (-1)^n*(n+3)!/(2*(n+1)) for n >= 0.
2

%I #15 Dec 18 2017 22:47:56

%S 3,-6,20,-90,504,-3360,25920,-226800,2217600,-23950080,283046400,

%T -3632428800,50295168000,-747242496000,11856247603200,

%U -200074178304000,3577797070848000,-67580611338240000,1344498478202880000,-28100018194440192000,615524208068689920000

%N a(n) = (-1)^n*(n+3)!/(2*(n+1)) for n >= 0.

%C Third column of Sheffer type triangle A238363 (without leading zeros).

%H Michael De Vlieger, <a href="/A238474/b238474.txt">Table of n, a(n) for n = 0..448</a>

%F a(n) = (-1)^n*(n+3)!/(2*(n+1)) for n >= 0.

%F E.g.f.: (d^3/dx^3)(log(1+x)*x^2/2!) = (x^2+3*x+3)/(1+x)^3.

%t Fold[Append[#1, (-1)^#2 (#2 + 3)!/(2 (#2 + 1))] &, {3}, Range@ 20] (* _Michael De Vlieger_, Dec 18 2017 *)

%Y Cf. A238363.

%K sign,easy

%O 0,1

%A _Wolfdieter Lang_, Mar 01 2014