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

1,5

Comments

Row sums are {0, 0, 0, -15, 1, -399, -399, -14399, -78399, -639999, -12959999}.

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

Table of n, a(n) for n=1..74.

G. B. Shabat and A. Zvonkin, Plane trees and algebraic numbers, Contemporary Math., 1994, vol. 178, 233-275.

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.

Example

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},
  ...

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]

Prog

(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

Crossrefs

Cf. A123583, A137286, A136247.

Sequence in context: A229694 A033596 A063529 * A004588 A272474 A340426

Adjacent sequences: A136664 A136665 A136666 * A136668 A136669 A136670

Keyword

uned,tabf,sign

Author

Roger L. Bagula, Apr 02 2008

Extensions

Keyword changed to tabf by Michel Marcus, Feb 26 2018

Status

approved