A203242 - OEIS

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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

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OFFSET

2,1

LINKS

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

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)

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 *)

CROSSREFS

Cf. A203241.