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 A195350 Expansion of (1 - 3*x - x^2)/(1 - 4*x + 2*x^3 + x^4). 10
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 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 TableGW.n=3 p. 22. Index entries for linear recurrences with constant coefficients, signature (4,0,-2,-1). 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 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 *) PROG (PARI) Vec((1-3*x-x^2)/(1-4*x+2*x^3+x^4)+O(x^26)) (MAGMA) m:=26; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-3*x-x^2)/(1-4*x+2*x^3+x^4))); (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); 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. Sequence in context: A164048 A180717 A151049 * A289810 A149044 A105284 Adjacent sequences:  A195347 A195348 A195349 * A195351 A195352 A195353 KEYWORD nonn,easy AUTHOR Bruno Berselli, Sep 16 2011 STATUS approved

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Last modified April 18 10:38 EDT 2019. Contains 322209 sequences. (Running on oeis4.)