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A203239 Odd numbered terms of the sequence whose n-th term is the (n-1)-st elementary symmetric function of (i, 2i, 3i, ..., ni), where i=sqrt(-1).

%I A203239 #26 Oct 26 2016 18:52:18
%S A203239 3,-50,1764,-109584,10628640,-1486442880,283465647360,-70734282393600,
%T A203239 22376988058521600,-8752948036761600000,4148476779335454720000,
%U A203239 -2342787216398718566400000,1554454559147562279567360000
%N A203239 Odd numbered terms of the sequence whose n-th term is the (n-1)-st elementary symmetric function of (i, 2i, 3i, ..., ni), where i=sqrt(-1).
%F A203239 a(n) = (-1)^(n+1)*(2*n)!*Sum_{i=1..2n} 1/i. - _Arkadiusz Wesolowski_, Mar 25 2013
%F A203239 From _Anton Zakharov_, Oct 26 2016: (Start)
%F A203239 a(n) = (-1)^(n+1)*Sum_{k=1..n} A094310(2n,k).
%F A203239 (-1)^(n+1)*a(n) = A000254(2n) (signed bisection of A000254). (End)
%e A203239 The first 10 terms of the "full sequence" are as follows:
%e A203239 1, 3i, -11, -50i, 274, 1764i, -13068, -109584i, 1026576, 10628640i;
%e A203239 Abbreviate "elementary symmetric function" as esf. Then, starting with {i, 2i, 3i, 4i, ...}:
%e A203239 0th esf of {i}: 1
%e A203239 1st esf of {i, 2i}: i+2i = 3i
%e A203239 2nd esf of {i, 2i, 3i}: -2-3-6 = -11.
%e A203239 For the alternating terms 3i, -50i, ..., see A203240.
%t A203239 f[k_] := k*I; t[n_] := Table[f[k], {k, 1, n}]
%t A203239 a[n_] := SymmetricPolynomial[n - 1, t[n]]
%t A203239 Table[a[n], {n, 1, 22}]
%t A203239 Table[-I*a[2 n], {n, 1, 22}]     (* A203239 *)
%t A203239 Table[a[2 n - 1], {n, 1, 22}]    (* A203240 *)
%t A203239 Table[(-1)^(n + 1)*(2*n)!*HarmonicNumber[2*n], {n, 13}] (* _Arkadiusz Wesolowski_, Mar 25 2013 *)
%Y A203239 Cf. A000254, A094310, A165675, A203240.
%K A203239 sign
%O A203239 1,1
%A A203239 _Clark Kimberling_, Dec 30 2011