A220666 Array of coefficients of powers of x^2 for S(2*n,x)^3 with Chebyshev's S polynomials A049310.

1, -1, 3, -3, 1, 1, -9, 30, -45, 30, -9, 1, -1, 18, -123, 399, -651, 588, -308, 93, -15, 1, 1, -30, 345, -1921, 5598, -9540, 10212, -7137, 3303, -1003, 192, -21, 1, -1, 45, -780, 6609, -29847, 80718, -141482, 168927, -141636, 84766, -36366, 11091, -2346, 327, -27, 1

Offset: 0

Author

Wolfdieter Lang, Dec 17 2012

Keywords

• sign
• easy
• tabf

Comments

The row lengths sequence of this array is 3*n+1 = A016777(n).

For the coefficient array of S(n,x)^3 see A219240. The present array is the even part of the bisection of that one.

The row polynomials in powers of x^2 are (S(2*n,x))^3 =

sum(a(n,m)*x^(2*m), m=0..3*n), n >= 0. The o.g.f. for these row polynomials is GS3even(x,z) = ((z+1)^3 + (1+z)*z*x^2*(3*x^2 - 7))/(((z+1)^2-z*x^2)*((z+1)^2-z*x^2*(x^2-3)^2)). This is obtained from the even part of the bisection of the o.g.f. for A219240.

Examples

			The array a(n,m) begins:
n\m  0    1     2     3      4     5     6    7    8  9
0:   1
1:  -1    3    -3     1
2:   1   -9    30   -45     30    -9     1
3:   1   18  -123   399   -651   588  -308   93  -15  1
...
Row n=4: [1, -30, 345, -1921, 5598, -9540, 10212, -7137, 3303, -1003, 192, -21, 1],
Row n=5: [-1, 45, -780, 6609, -29847, 80718, -141482, 168927, -141636, 84766, -36366, 11091, -2346, 327, -27, 1],
Row n=6: [1, -63, 1533, -18333, 118029, -460815, 1184872, -2118207, 2729922, -2598297, 1854177, -999687, 407472, -124680, 28164, -4553, 498, -33, 1].
Row n=2: S(4,x)^3 = 1 - 9*x^2 + 30*x^4 - 45*x^6 + 30*x^8  - 9*x^10 + 1*x^12.
		

Crossrefs

Cf. A219240, A220665 (odd part of the bisection).

Formula

a(n,m) = [x^m] S(2*n,x)^3, n>=0, 0 <= m <= 3*n.

a(n,m) = [x^m]([z^n]GS3even(x,z)) with GS3even(x,z) the o.g.f. for the row polynomials in powers of x^2, given in a comment above.