A301420 Sums of positive coefficients in generalized Chebyshev polynomials of the first kind, for a family of 5 data.

1, 5, 31, 205, 1376, 9251, 62210, 418361, 2813485, 18920751, 127242501, 855708865, 5754662616, 38700243965, 260260067876, 1750255192001, 11770508100345, 79156948982921, 532332378421395, 3579947998967501, 24075236064574376

Offset: 1

Author

Gregory Gerard Wojnar, Mar 20 2018

Keywords

• nonn

Comments

Re-express the Girard-Waring formulae to yield the mean powers in terms of the mean symmetric polynomials in the data values. Then for a family of 5 data, the sum of the positive coefficients in these polynomials is a(n). a(n+1)/a(n) approaches 1/(2^(1/5)-1). (For a family of 2 data, the coefficients of these polynomials give the Chebyshev polynomials of the first kind.) See extended comment in A301417.

Links

• Gregory Gerard Wojnar, Table of n, a(n) for n = 1..65
• 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=5 p. 23.

Crossrefs

Cf. A302764, A024537, A195350, A301417, A301421, A301424.

Programs

PARI

lista(5, nn) \\ use pari script file in A301417; Michel Marcus, Apr 21 2018

Formula

G.f.: (-x*(x+1)^4+1)/(x^6+4*x^5+5*x^4-5*x^2-6*x+1); this denominator equals (1-x)*(2-(x+1)^5) (conjectured).

a(n+14) = 7*a(n+13) - a(n+12) - 6*a(n+11) + 2*a(n+10) - a(n+9) + 4*a(n+8) + a(n+7) + 4*a(n+5) + 2*a(n+4) - a(n+3) - 5*a(n+2) - 4*a(n+1) - a(n) (conjectured).