A219234 Coefficient array for the fourth power of Chebyshev's S-polynomials as a function of x^2.

1, 0, 0, 1, 1, -4, 6, -4, 1, 0, 0, 16, -32, 24, -8, 1, 1, -12, 58, -144, 195, -144, 58, -12, 1, 0, 0, 81, -432, 972, -1200, 886, -400, 108, -16, 1, 1, -24, 236, -1228, 3678, -6612, 7490, -5532, 2701, -864, 174, -20, 1, 0, 0, 256, -2560, 11136, -27776, 44176, -47232, 34912, -18048, 6504, -1600, 256, -24, 1

Offset: 0

Author

Wolfdieter Lang, Nov 28 2012

Keywords

• sign
• easy
• tabf

Comments

The row lengths sequence for this array is 2*n+1, given in A005408.

The coefficient triangle for the monic Chebyshev S-polynomials S(n,x) = U(n,x/2) are given in A049310.

The coefficients for S(n,x)^2 are given in A158454 and in A181878 (odd numbered rows shifted by one unit to the left).

Examples

			The irregular triangle a(n, m) starts:
n\m  0   1   2     3    4     5    6     7    8    9  10  11 12
0:   1
1:   0   0   1
2:   1  -4   6    -4    1
3:   0   0  16   -32   24    -8    1
4:   1 -12  58  -144  195  -144   58   -12    1
5:   0   0  81  -432  972 -1200  886  -400  108  -16   1
6:   1 -24 236 -1228 3678 -6612 7490 -5532 2701 -864 174 -20  1
...
Row n=7: [0, 0, 256, -2560, 11136, -27776, 44176, -47232, 34912, -18048, 6504, -1600, 256, -24, 1].
Row n=8: [1, -40, 660, -5828, 30194, -96780, 203374, -293464, 300231, -222112, 119938, -47244, 13415, -2672, 354, -28, 1].
Row n=1 polynomial p(1,x) = 1*x^2 = S(1,sqrt(x))^4 = (sqrt(x))^4.
Row n=2 polynomial p(2,x) = 1 - 4*x + 6*x^2 - 4*x^3 + 1*x^4 =
  S(2,sqrt(x))^4 = (-1+x)^4.
		

Crossrefs

Cf. A049310, A127672, A158454, A181878, A219240.

Formula

a(n, m) = [x^(2*m)] S(n, x)^4, n >= 0, with the monic Chebyshev S-polynomials given in terms of the U-polynomials in a comment above.

The o.g.f. GS4(x, z) := sum((S(n, x)^4)*z^n,n=0..infinity) = ((1+z)/(1-z))*(1 - (2-3*x^2)*z + z^2)/((1-z*(-2+x^2)+z^2)*(1-z*(2-4*x^2+x^4)+z^2)). For the o.g.f. of the row polynomials p(n,x) :=sum(a(n,m)*x^m,m=0..n) take GS4(sqrt(x), z).

The row polynomial p(n, x^2) = Sum_{m=0..2*n} a(n, m)*x^(2*m) = (S(n, x))^4 = (R(4*(n+1), x) - 4*R(2*(n+1), x) + 6)/(x^2 - 4)^2, where R are the monic Chebyshev T polynomials with coefficients given in A127672. For factorizations of the S polynomials see comments on A049310. - Wolfdieter Lang, Apr 09 2018