A217476 Coefficient triangle for the square of the monic integer Chebyshev T-polynomials A127672.

4, 0, 1, 4, -4, 1, 0, 9, -6, 1, 4, -16, 20, -8, 1, 0, 25, -50, 35, -10, 1, 4, -36, 105, -112, 54, -12, 1, 0, 49, -196, 294, -210, 77, -14, 1, 4, -64, 336, -672, 660, -352, 104, -16, 1, 0, 81, -540, 1386, -1782, 1287, -546, 135, -18, 1, 4, -100, 825, -2640, 4290, -4004, 2275, -800, 170, -20, 1

Offset: 0

Author

Wolfdieter Lang, Oct 17 2012

Keywords

• sign
• easy
• tabl

Comments

The monic integer T-polynomials, called R(n,x) (in Abramowitz-Stegun C(n,x)), with their coefficient triangle given in A127672, when squared, become polynomials in y=x^2:

R(n,x)^2 = sum(T(n,k)*y^k,m=0..n).

R(n,x)^2 = 2 + R(2*n,x). From the bisection of the R-(or T-)polynomials, the even part. Directly from the R(m*n,x)=R(m,R(n,x)) property for m=2.

The o.g.f. is G(z,y) := sum((R(n,sqrt(y))^2)*z^n ,n=0..infinity) = (4 + (4 - 3*y)*z + y*z^2)/((1 +(2-y)*z + z^2)*(1-z)). From the bisection.

The o.g.f.s of the columns k>=1 are x^k*(1-x)/(1+x)^(2*k+1),

and for k=0 the o.g.f. is 4/(1-x^2).

Hetmaniok et al. (2015) refer to these as "modified Chebyshev" polynomials. - N. J. A. Sloane, Sep 13 2016

Examples

			The triangle begins:
n\k 0    1    2      3     4      5     6     7    8   9  10
0:  4
1:  0    1
2:  4   -4    1
3:  0    9   -6      1
4:  4  -16   20     -8     1
5:  0   25  -50     35   -10      1
6:  4  -36  105   -112    54    -12     1
7:  0   49 -196    294  -210     77   -14     1
8:  4  -64  336   -672   660   -352   104   -16    1
9:  0   81 -540   1386 -1782   1287  -546   135  -18   1
10: 4 -100  825  -2640  4290  -4004  2275  -800  170 -20   1
...
n=2:  R(2,x) = -2 + y, R(2,x)^2 = 4 -4*y + y^2, with y=x^2.
n=3:  R(3,x) = 3*x - x^3, R(3,x)^2 = 9*y - 6*y^2 +y^3, with y=x^2.
T(4,1) = 8*(-1)^3*binomial(5,3)/5 = -16.
T(4,0) = 2 + 8*(-1)^4*binomial(4,4)/4 = 4.
T(n,1) = (-1)^(n-1)*2*n*(n+1)!/((n-1)!*2!*(n+1)) = -((-1)^n)*n^2 = A162395(n), n >= 1.
T(n,2) = (-1)^n*A002415(n), n >= 0.
T(n,3) = -(-1)^n*A040977(n-3), n >= 3.
T(n,4) = (-1)^n*A053347(n-4), n >= 4.
T(n,5) = -(-1)^n*A054334(n-5), n >= 5.
		

References

• E Hetmaniok, P Lorenc, S Damian, et al., Periodic orbits of boundary logistic map and new kind of modified Chebyshev polynomials in R. Witula, D. Slota, W. Holubowski (eds.), Monograph on the Occasion of 100th Birthday Anniversary of Zygmunt Zahorski. Wydawnictwo Politechniki Slaskiej, Gliwice 2015, pp. 325-343.

Crossrefs

Cf. A127672, A158454 (square of S-polynomials), A128495 (sum of square of S-polynomials).

Formula

T(n,k) = [x^(2*k)]R(n,x)^2, with R(n,x) the monic integer version of the Chebyshev T(n,x) polynomial.

T(n,k) = 0 if n=1. ([k=0] means 1 if k=0 else 0).