A301421 Sums of positive coefficients of generalized Chebyshev polynomials of the first kind, for a family of 6 data.

1, 6, 46, 371, 3026, 24707, 201748, 1647429, 13452565, 109850886, 897019828, 7324880157, 59813470848, 488424550081, 3988374821616, 32568251770049, 265945672309613, 2171657880797162, 17733313387923690, 144806604435722311, 1182461068019218530, 9655734852907204771

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 6 data, the sum of the positive coefficients in these polynomials is a(n). a(n+1)/a(n) approaches 1/(2^(1/6)-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..62 [a(21) corrected by _Georg Fischer_, Aug 18 2021]
• 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=6 p. 24.

Crossrefs

Cf. A301764, A024537, A195350, A301417, A301420, A301424.

Programs

PARI

lista(6, nn) \\ use pari script file in A301417; Michel Marcus, Apr 21 2018

Formula

G.f.: (-x*(x+1)^5+1)/(x^7+5*x^6+9*x^5+5*x^4-5*x^3-9*x^2-7*x+1); this denominator equals (1-x)*(2-(1+x)^6) (conjectured).

Extensions

a(21) corrected by Georg Fischer, Aug 18 2021