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A203246 Second elementary symmetric function of the first n terms of (1,1,2,2,3,3,4,4,...).

%I A203246 #25 Dec 22 2024 21:32:34
%S A203246 1,5,13,31,58,106,170,270,395,575,791,1085,1428,1876,2388,3036,3765,
%T A203246 4665,5665,6875,8206,9790,11518,13546,15743,18291,21035,24185,27560,
%U A203246 31400,35496,40120,45033,50541,56373,62871,69730,77330,85330,94150,103411,113575
%N A203246 Second elementary symmetric function of the first n terms of (1,1,2,2,3,3,4,4,...).
%C A203246 Second subdiagonal of A246117. - _Peter Bala_, Aug 15 2014
%H A203246 Sela Fried, <a href="/A203246/a203246.pdf">On A203246</a>, 2024.
%H A203246 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-6,0,6,-2,-2,1).
%F A203246 Conjectural o.g.f.: x^2*(1 + 3*x + x^2 + x^3)/((1 - x^2)^3*(1 - x)^2). - _Peter Bala_, Aug 15 2014
%F A203246 Conjectural closed form: 64*a(n) = 2*n^2 -16*n/3 -3 +16*n^3/3 +2*n^4 +(-1)^n *(3-2*n^2). - _R. J. Mathar_, Oct 01 2016
%F A203246 Both conjectures are true. See link. - _Sela Fried_, Dec 22 2024
%t A203246 f[k_] := Floor[(k + 1)/2]; t[n_] := Table[f[k], {k, 1, n}]
%t A203246 a[n_] := SymmetricPolynomial[2, t[n]]
%t A203246 Table[a[n], {n, 2, 50}]  (* A203246 *)
%Y A203246 Cf. A203298, A203299, A246117, A212523 (odd bisection), A103220 (even bisection).
%K A203246 nonn,easy
%O A203246 2,2
%A A203246 _Clark Kimberling_, Dec 31 2011