A301424 Sums of positive coefficients of generalized Chebyshev polynomials of the first kind, for a family of 7 data.

1, 7, 64, 609, 5846, 56161, 539540, 5183417, 49797685, 478412117, 4596160548, 44155846113, 424210322004, 4075437640457, 39153200900024, 376149330687809, 3613710136705565, 34717331354145139, 333533418773956668, 3204294140706218329, 30784024515164777522

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 7 data, the sum of the positive coefficients in these polynomials is a(n). a(n+1)/a(n) approaches 1/(2^(1/7)-1). (For a family of 2 data, the coefficients of these polynomials give the Chebyshev polynomials of the first kind.) The sums of the negative coefficients are 1 less than the corresponding sums of the positive coefficients. See extended comment in A301417.

Links

• G. G. Wojnar, D. S. Wojnar, and L. Q. Brin, Universal peculiar linear mean relationships in all polynomials, arXiv:1706.08381 [math.GM], 2017.

Crossrefs

Cf. A302764, A024537, A195350, A301417, A301420, A301421.

Programs

PARI

lista(7, nn) \\ use pari script file in A301417; Michel Marcus, Apr 21 2018

Formula

G.f.: (-x*(x+1)^6+1)/(x^2*(x^6+6*x^5+14*x^4+14*x^3-14*x-14)-8*x+1); this denominator equals (1-x)*(2-(1+x)^7) (conjectured).