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

1, 5, 13, 31, 58, 106, 170, 270, 395, 575, 791, 1085, 1428, 1876, 2388, 3036, 3765, 4665, 5665, 6875, 8206, 9790, 11518, 13546, 15743, 18291, 21035, 24185, 27560, 31400, 35496, 40120, 45033, 50541, 56373, 62871, 69730, 77330, 85330, 94150, 103411, 113575

Offset: 2

Author

Clark Kimberling, Dec 31 2011

Keywords

• nonn
• easy

Comments

Second subdiagonal of A246117. - Peter Bala, Aug 15 2014

Links

• Sela Fried, On A203246, 2024.
• Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Crossrefs

Cf. A203298, A203299, A246117, A212523 (odd bisection), A103220 (even bisection).

Programs

Mathematica

f[k_] := Floor[(k + 1)/2]; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[2, t[n]]
Table[a[n], {n, 2, 50}]  (* A203246 *)

Formula

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

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

Both conjectures are true. See link. - Sela Fried, Dec 22 2024