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%I #13 Jan 04 2018 09:03:56
%S 2,8,23,47,91,151,246,366,540,750,1037,1373,1813,2317,2956,3676,4566,
%T 5556,6755,8075,9647,11363,13378,15562,18096,20826,23961,27321,31145,
%U 35225,39832,44728,50218,56032,62511,69351,76931,84911,93710,102950
%N Second elementary symmetric function of the first n terms of (1,2,2,3,3,4,4,5,5...).
%F Empirical g.f.: x^2*(2 + 4*x + 3*x^2 - 3*x^3 - x^4 + x^5) / ((1 - x)^5*(1 + x)^3). - _Colin Barker_, Aug 15 2014
%F Conjectures from _Colin Barker_, Jan 04 2018: (Start)
%F a(n) = (6*n^4 + 40*n^3 + 48*n^2 - 112*n) / 192 for n even.
%F a(n) = (6*n^4 + 40*n^3 + 36*n^2 - 136*n + 54) / 192 for n odd.
%F a(n) = 2*a(n-1) + 2*a(n-2) - 6*a(n-3) + 6*a(n-5) - 2*a(n-6) - 2*a(n-7) + a(n-8) for n>9.
%F (End)
%t f[k_] := Floor[(k + 2)/2]; t[n_] := Table[f[k], {k, 1, n}]
%t a[n_] := SymmetricPolynomial[2, t[n]]
%t Table[a[n], {n, 2, 50}] (* A203298 *)
%Y Cf. A203246, A203299.
%K nonn
%O 2,1
%A _Clark Kimberling_, Dec 31 2011
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