login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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
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 (list; graph; refs; listen; history; text; internal format)
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 A308717

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

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.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 21 19:39 EDT 2019. Contains 327279 sequences. (Running on oeis4.)