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A203423 a(n) = w(n+1)/(2*w(n)), where w=A203422.

%I A203423 #25 Dec 08 2023 12:44:40
%S A203423 -3,24,-250,3240,-50421,917504,-19131876,450000000,-11789738455,
%T A203423 340545503232,-10752962364222,368510430439424,-13623365478515625,
%U A203423 540431955284459520,-22899384412078526344,1032236014321051140096,-49323481720063219673451,2490368000000000000000000,-132484966403310261255807810
%N A203423 a(n) = w(n+1)/(2*w(n)), where w=A203422.
%H A203423 G. C. Greubel, <a href="/A203423/b203423.txt">Table of n, a(n) for n = 1..350</a>
%F A203423 a(n) = (-1)^n*A053506(n+2)/2. - _Steven Finch_, Apr 16 2022
%F A203423 E.g.f.: -(1/(2*x^2))*( W(x)/(1 + W(x))^3 - 2*W(x)/(1 + W(x)) + W(x) + x^2), where W(x) = LambertW(x). - _G. C. Greubel_, Dec 07 2023
%t A203423 (* First program *)
%t A203423 f[j_] := 1/(j + 1); z = 16;
%t A203423 v[n_] := Product[Product[f[k] - f[j], {j, 1, k - 1}], {k, 2, n}]
%t A203423 1/Table[v[n], {n, 1, z}]             (* A203422 *)
%t A203423 Table[v[n]/(2 v[n + 1]), {n, 1, z}]  (* this sequence *)
%t A203423 (* Second program *)
%t A203423 Table[(-1)^n*(n+1)*(n+2)^n/2, {n,20}] (* _G. C. Greubel_, Dec 07 2023 *)
%o A203423 (Magma) [(-1)^n*(n+1)*(n+2)^n/2: n in [1..20]]; // _G. C. Greubel_, Dec 07 2023
%o A203423 (SageMath) [(-1)^n*(n+1)*(n+2)^n/2 for n in range(1,21)] # _G. C. Greubel_, Dec 07 2023
%Y A203423 Cf. A000169, A053506, A203421, A203422.
%K A203423 sign
%O A203423 1,1
%A A203423 _Clark Kimberling_, Jan 02 2012