A195350 Expansion of (1 - 3*x - x^2)/(1 - 4*x + 2*x^3 + x^4).

1, 1, 3, 10, 37, 141, 541, 2080, 8001, 30781, 118423, 455610, 1752877, 6743881, 25945881, 99822160, 384048001, 1477556361, 5684635243, 21870622810, 84143330517, 323726495221, 1245480100021, 4791763116240, 18435456144001, 70927137880741

Offset: 0

Author

Bruno Berselli, Sep 16 2011

Keywords

• nonn
• easy

Comments

Rewrite the Girard-Waring formulae to express the mean powers in terms of the mean symmetric functions of the data values; the results are polynomials in the mean symmetric polynomials, indexed by the power n. Then for 3 data points, the sum of the positive coefficients in the n-th such polynomial is a(n). a(n+1)/a(n) approaches 1/(2^(1/3)-1). See extended comment in A301417. - Gregory Gerard Wojnar, Mar 19 2018

Links

• Bruno Berselli, Table of n, a(n) for n = 0..1000
• G. G. Wojnar, D. S. Wojnar, and L. Q. Brin, Universal peculiar linear mean relationships in all polynomials, arXiv:1706.08381 [math.GM], 2017. See Table GW. n=3 p. 22.
• Index entries for linear recurrences with constant coefficients, signature (4,0,-2,-1).

Crossrefs

Cf. A185962 (gives the coefficients of numerator and denominator of the g.f., row 4 and 5 of its triangular array). Sequences likewise related to A185962: A000012 (row 1 and 2), A001333 (row 2 and 3) and A006190 (row 3 and 4).

Cf. also A195339, A301417, A301420, A301421, A301424, A302764, A301483, A303647.

Programs

Magma

m:=26; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-3*x-x^2)/(1-4*x+2*x^3+x^4)));

Maple

[seq(coeftayl((1-3*x-x^2)/(1-4*x+2*x^3+x^4), x = 0, k), k=0..25)]; # Muniru A Asiru, Mar 20 2018

Mathematica

CoefficientList[Series[(1 - 3 x - x^2)/(1 - 4 x + 2 x^3 + x^4), {x, 0, 25}], x] (* Vincenzo Librandi, Mar 26 2013 *)

Maxima

makelist(coeff(taylor((1-3*x-x^2)/(1-4*x+2*x^3+x^4), x, 0, n), x, n), n, 0, 25);

PARI

Vec((1-3*x-x^2)/(1-4*x+2*x^3+x^4)+O(x^26))

Formula

G.f.: (1-3*x-x^2)/((1-x)*(1-3*x-3*x^2-x^3)).

a(n) = 4*a(n-1) - 2*a(n-3) - a(n-4).

a(n) = A301483(n) - A303647(n-2) + A195339(n-4) (conjectured). - Gregory Gerard Wojnar, Apr 27 2018