A203242 Second elementary symmetric function of the first n terms of (1, 3, 7, 15, 31, ...).

3, 31, 196, 1002, 4593, 19833, 82818, 339340, 1375639, 5543331, 22263216, 89249214, 357422541, 1430607325, 5724394990, 22901773824, 91616007699, 366482904615, 1465971463740, 5863969740370, 23456055121513, 93824589584001

Offset: 2

Author

Clark Kimberling, Dec 31 2011

Keywords

• nonn
• easy

Links

• Robert Israel, Table of n, a(n) for n = 2..1659

Crossrefs

Cf. A203241.

Programs

Maple

seq(4^(n+1)/3 - (2*n+2)*2^n + (n^2+3*n)/2 + 2/3,n=2..100); # Robert Israel, Feb 01 2019

Mathematica

f[k_] := 2^k - 1; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[2, t[n]]
Table[a[n], {n, 2, 32}]  (* A203242 *)

Formula

Conjecture: (-103*n+258)*a(n) + (881*n-2116)*a(n-1) + 6*(-427*n+960)*a(n-2) + 4*(766*n-1545)*a(n-3) + 16*(-80*n+121)*a(n-4) = 0. - R. J. Mathar, Oct 15 2013

Empirical g.f.: -x^2*(4*x^2 + 2*x - 3)/((x - 1)^3*(2*x - 1)^2*(4*x - 1)). - Colin Barker, Aug 15 2014

From Robert Israel, Feb 01 2019: (Start)

Conjecture and empirical g.f. verified.

a(n) = 4^(n+1)/3 - (2*n+2)*2^n + (n^2+3*n)/2 + 2/3. (End)