This site is supported by donations to The OEIS Foundation.

Thanks to everyone who made a donation during our annual appeal!
To see the list of donors, or make a donation, see the OEIS Foundation home page.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A113279 Triangle T(n,k) of coefficients of r_1^n+r_2^n in terms of p and q, where r_1,r_2 are the roots of x^2+px+q=0. 3
 2, -1, 1, -2, -1, 3, 1, -4, 2, -1, 5, -5, 1, -6, 9, -2, -1, 7, -14, 7, 1, -8, 20, -16, 2, -1, 9, -27, 30, -9, 1, -10, 35, -50, 25, -2, -1, 11, -44, 77, -55, 11, 1, -12, 54, -112, 105, -36, 2, -1, 13, -65, 156, -182, 91, -13, 1, -14, 77, -210, 294, -196, 49, -2, -1, 15, -90, 275, -450, 378, -140, 15 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS The triangle begins: 2 .-1 1..-2 .-1..3 1..-4.2 From Tom Copeland, Nov 07 2015: (Start) The row polynomials are the power sums p_n = x1^n + x2^n of the solutions of 0 = (x-x1)(x-x2) = x^2 - (x1+x2) x + x1x2 = x^2 + px + q, so -p = x1+x2 = e1(x1,x2), the first order elementary symmetric polynomial for two variables, or indeterminates, and q = x1*x2 = e2(x1,x2), the second order elementary symmetric polynomial. The Girard-Newton-Waring identities (Wikipedia) express the power sums in terms of the elementary symmetric polynomials, giving p_0 = x1^0 + x0^0 = 2 p_1 = x1 + x2 = e1 = -p = F(1,p,q,0,..) p_2 = x1^2 + x2^2 =  e1^2 - 2 e2 = p^2 - 2 q = F(2,p,q,0,..) p_3 = e1^3 - 3 e2 e1 = -p^3 + 3 pq = F(3,p,q,0,..) p_4 = p^4 - 4 p^2 q + 2 q^2 = F(4,p,q,0,..) ... . These bivariate partition polynomials are the Faber polynomials F(n,b1,b2,..,bn) of A263916 with b1 = -e1 = p, b2 = e2 = q, and all other indeterminates set to zero. Let p = q = t, then F(1,t,t,0,..)/t = -1 F(2,t,t,0,..)/t = -2 + t F(3,t,t,0,..)/t^2 = 3 - t F(4,t,t,0,..)/t^2 = 2 - 4 t + t^2 ... . Or, t * F(1,1/t,1/t,0,..) = -1 t^2 * F(2,1/t,1/t,0,..) = 1 -2 t t^3 * F(3,1/t,1/t,0,..) = -1 + 3 t t^4 * F(4,1/t,1/t,0,..) = 1 - 4 t + 2 t^2 ... . The sequence of Faber polynomials F(n,1/t,1/t,0,..) is obtained from the logarithmic generator -log[1+(x+x^2)/t] = sum{n>=1, F(n,1/t,1/t,0,..) x^n/n}, so   2-(2xt+1)xt/(t+xt+(xt)^2) = 2 + sum{n>=1, t^n F(n,1//t,1/t,0,..) x^n} is an o.g.f. for the row polynomials of this entry. (End) REFERENCES M. Herkenhoff Konersmann, Sprokkel XXXI: x_1^n+x_2^n, Nieuw Tijdschr. Wisk, 42 (1954-55) 180. LINKS T. Copeland, Addendum to Elliptic Lie Triad Wikipedia, Newton's identities. FORMULA T(n, k) = (-1)^(n+k)*A034807(n, k). O.g.f.: 2-(2xt+1)xt/(t+xt+(xt)^2) = (2+x)/(1+x+x^2/t). - Tom Copeland, Nov 07 2015 EXAMPLE x_1^5+x_2^5 = -p^5 + 5p^3q - 5pq^2, so row 5 reads -1, 5, -5. CROSSREFS Cf. A034807, A263916. Sequence in context: A055893 A050221 A213235 * A213234 A034807 A275111 Adjacent sequences:  A113276 A113277 A113278 * A113280 A113281 A113282 KEYWORD easy,sign,tabf AUTHOR Floor van Lamoen, Oct 22 2005 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified January 16 19:49 EST 2019. Contains 319206 sequences. (Running on oeis4.)