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).
3, -50, 1764, -109584, 10628640, -1486442880, 283465647360, -70734282393600, 22376988058521600, -8752948036761600000, 4148476779335454720000, -2342787216398718566400000, 1554454559147562279567360000
Offset: 1
Keywords
Examples
The first 10 terms of the "full sequence" are as follows:
1, 3i, -11, -50i, 274, 1764i, -13068, -109584i, 1026576, 10628640i;
Abbreviate "elementary symmetric function" as esf. Then, starting with {i, 2i, 3i, 4i, ...}:
0th esf of {i}: 1
1st esf of {i, 2i}: i+2i = 3i
2nd esf of {i, 2i, 3i}: -2-3-6 = -11.
For the alternating terms 3i, -50i, ..., see A203240.
Programs
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Mathematica
f[k_] := k*I; t[n_] := Table[f[k], {k, 1, n}] a[n_] := SymmetricPolynomial[n - 1, t[n]] Table[a[n], {n, 1, 22}] Table[-I*a[2 n], {n, 1, 22}] (* A203239 *) Table[a[2 n - 1], {n, 1, 22}] (* A203240 *) Table[(-1)^(n + 1)*(2*n)!*HarmonicNumber[2*n], {n, 13}] (* Arkadiusz Wesolowski, Mar 25 2013 *)
Formula
a(n) = (-1)^(n+1)*(2*n)!*Sum_{i=1..2n} 1/i. - Arkadiusz Wesolowski, Mar 25 2013
From Anton Zakharov, Oct 26 2016: (Start)
a(n) = (-1)^(n+1)*Sum_{k=1..n} A094310(2n,k).