A203298 Second elementary symmetric function of the first n terms of (1,2,2,3,3,4,4,5,5...).

2, 8, 23, 47, 91, 151, 246, 366, 540, 750, 1037, 1373, 1813, 2317, 2956, 3676, 4566, 5556, 6755, 8075, 9647, 11363, 13378, 15562, 18096, 20826, 23961, 27321, 31145, 35225, 39832, 44728, 50218, 56032, 62511, 69351, 76931, 84911, 93710, 102950

Offset: 2

Author

Clark Kimberling, Dec 31 2011

Keywords

• nonn

Crossrefs

Cf. A203246, A203299.

Programs

Mathematica

f[k_] := Floor[(k + 2)/2]; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[2, t[n]]
Table[a[n], {n, 2, 50}]  (* A203298 *)

Formula

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

Conjectures from Colin Barker, Jan 04 2018: (Start)

a(n) = (6*n^4 + 40*n^3 + 48*n^2 - 112*n) / 192 for n even.

a(n) = (6*n^4 + 40*n^3 + 36*n^2 - 136*n + 54) / 192 for n odd.

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.

(End)