A220668 Coefficient array for the powers of x^2 of the square of the even-indexed Chebyshev C polynomials.

4, 4, -4, 1, 4, -16, 20, -8, 1, 4, -36, 105, -112, 54, -12, 1, 4, -64, 336, -672, 660, -352, 104, -16, 1, 4, -100, 825, -2640, 4290, -4004, 2275, -800, 170, -20, 1, 4, -144, 1716, -8008, 19305, -27456, 24752, -14688, 5814, -1520, 252, -24, 1, 4, -196, 3185, -20384, 68068, -136136, 176358, -155040, 94962, -40964, 12397, -2576, 350, -28, 1

Offset

0,1

Comments

The row lengths sequence of this irregular triangle is 2*n + 1 = A005408(n), n>=0.

For the coefficient triangle for Chebyshev's C polynomials see A127672 (where they are called R polynomials).

a(n,m) is the coefficient of (x^2)^m of C(2*n,x)^2. The o.g.f. for the row polynomials P(n,x) = sum(a(n,m)*x^m,m=0..2*n) is GC2even(x,z) := sum( P(n,x)*z^n,n=0..infinity) =

(4 - (8 - 12*x + 3*x^2)*z + (x - 2)^2*z^2)/((1 - z)*(1 - ((x-2)^2 - 2)*z + z^2)). From the even part of the bisection of the o.g.f. for the square of the C polynomials.

Links

Table of n, a(n) for n=0..63.

Formula

a(n,m) = [x^m] C(n,x)^2, n >= 0, 0 <= m <= 2*n, with Chebyshev's C polynomials (see A127672).

a(n,m) =[x^m]([z]^n GC2even(x,z)), with the o.g.f. GC2even(x,z) given in a comment above.

Example

The array begins:
n\m 0     1    2      3     4     5     6     7    8    9  10
0:  4
1:  4    -4    1
2:  4   -16   20     -8     1
3:  4   -36  105   -112    54   -12     1
4:  4   -64  336   -672   660  -352   104   -16    1
5:  4  -100  825  -2640  4290 -4004  2275  -800  170  -20   1
...
Row 6: [4, -144, 1716, -8008, 19305, -27456, 24752, -14688, 5814, -1520, 252, -24, 1],
Row 7: [4, -196, 3185, -20384, 68068, -136136, 176358, -155040, 94962, -40964, 12397, -2576, 350, -28, 1].
Row n=2: C(2,x)^2 = (-2 + x^2)^2 = 4 - 4*x^2 + 1*x^4, with
the row polynomial P(2,x) = C(2,sqrt(x))^2 = 4 - 4*x + 1*x^2.

Crossrefs

Cf. A127672.

Sequence in context: A159891 A201941 A180060 * A109610 A067395 A213274

Adjacent sequences: A220665 A220666 A220667 * A220669 A220670 A220671

Keyword

sign,easy,tabf

Author

Wolfdieter Lang, Dec 26 2012

Status

approved