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Expansion of (1 - 3*x - x^2)/(1 - 4*x + 2*x^3 + x^4).
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%I #64 Sep 08 2022 08:45:59

%S 1,1,3,10,37,141,541,2080,8001,30781,118423,455610,1752877,6743881,

%T 25945881,99822160,384048001,1477556361,5684635243,21870622810,

%U 84143330517,323726495221,1245480100021,4791763116240,18435456144001,70927137880741

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

%C 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

%H Bruno Berselli, <a href="/A195350/b195350.txt">Table of n, a(n) for n = 0..1000</a>

%H G. G. Wojnar, D. S. Wojnar, and L. Q. Brin, <a href="http://arxiv.org/abs/1706.08381">Universal peculiar linear mean relationships in all polynomials</a>, arXiv:1706.08381 [math.GM], 2017. See Table GW. n=3 p. 22.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,0,-2,-1).

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

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

%F a(n) = A301483(n) - A303647(n-2) + A195339(n-4) (conjectured). - _Gregory Gerard Wojnar_, Apr 27 2018

%p [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

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

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

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

%o (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);

%Y 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).

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

%K nonn,easy

%O 0,3

%A _Bruno Berselli_, Sep 16 2011