A136667 Triangle read by rows: T(n, k) is the coefficient of x^k in the polynomial 1 - T_n(x)^2, where T_n(x) is the n-th Hermite polynomial of the Hochstadt kind (A137286) as related to the generalized Chebyshev in a Shabat way (A123583): p(x,n)=x*p(x,n-1)-p(x,n-2); q(x,n)=1-p(x,n)^2.
0, 1, 0, -1, -3, 0, 4, 0, -1, 1, 0, -25, 0, 10, 0, -1, -63, 0, 144, 0, -97, 0, 18, 0, -1, 1, 0, -1089, 0, 924, 0, -262, 0, 28, 0, -1, -2303, 0, 8352, 0, -9489, 0, 3576, 0, -574, 0, 40, 0, -1, 1, 0, -77841, 0, 103230, 0, -49291, 0, 10548, 0, -1099, 0, 54, 0, -1, -147455, 0, 748800, 0, -1215585, 0, 699630, 0, -188043, 0
Offset: 1
Examples
The irregular triangle begins
{0},
{1, 0, -1},
{-3, 0, 4, 0, -1},
{1, 0, -25, 0, 10, 0, -1},
{-63, 0, 144, 0, -97, 0, 18, 0, -1},
{1, 0, -1089, 0, 924, 0, -262,0, 28, 0, -1},
{-2303, 0, 8352, 0, -9489, 0, 3576, 0, -574, 0, 40, 0, -1},
{1, 0, -77841, 0, 103230, 0, -49291, 0, 10548,0, -1099, 0, 54, 0, -1},
...
References
- Defined: page 8 and pages 42 - 43: Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986
- G. B. Shabat and I. A. Voevodskii, Drawing curves over number fields, The Grothendieck Festschift, vol. 3, Birkhäuser, 1990, pp. 199-22
Links
- G. B. Shabat and A. Zvonkin, Plane trees and algebraic numbers, Contemporary Math., 1994, vol. 178, 233-275.
Programs
-
Mathematica
P[x, 0] = 1; P[x, 1] = x; P[x_, n_] := P[x, n] = x*P[x, n - 1] - n*P[x, n - 2]; Q[x_, n_] := Q[x, n] = 1 - P[x, n]^2; Table[ExpandAll[Q[x, n]], {n, 0, 10}]; a = Join[{{0}}, Table[CoefficientList[Q[x, n], x], {n, 0, 10}]]; Flatten[a] -
PARI
polx(n) = if (n == 0, 1, if (n == 1, x, x*polx(n - 1) - n*polx(n - 2))); tabf(nn) = {for (n = 0, nn, pol = 1 - polx(n)^2; for (i = 0, 2*n, print1(polcoeff(pol, i), ", "); ); print(); ); } \\ Michel Marcus, Feb 26 2018
Formula
out=1-A137286(x,n)^2; p(x,n)=x*p(x,n-1)-p(x,n-2); q(x,n)=1-p(x,n)^2.
Extensions
Keyword changed to tabf by Michel Marcus, Feb 26 2018
Comments