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A220669 Coefficient array for powers of x^2 of the square of Chebyshev's C(2*n+1,x)/x =: tau(n,x) polynomials. 0
1, 9, -6, 1, 25, -50, 35, -10, 1, 49, -196, 294, -210, 77, -14, 1, 81, -540, 1386, -1782, 1287, -546, 135, -18, 1, 121, -1210, 4719, -9438, 11011, -8008, 3740, -1122, 209, -22, 1, 169, -2366, 13013, -37180, 63206, -68952, 50388, -25194, 8645, -2002, 299, -26, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

For the coefficient array of powers of x^2 of the monic integer Chebyshev C(2*n+1,x)/x = :tau(n,x) polynomials see the signed triangle ((-1)^(n-m))*A111125(n,m). See the comment from Oct 23 2012.

The o.g.f. of the row polynomials p(n,x) := sum(a(n,m)*x^m, m=0..2*n), n>=0, is G2(x,z) = sum(p(n,x)*z^n,n=0..infinity) =  (1+ (6-2*x)*z+z^2)/((1-z)*((z+1)^2-z*(x-2)^2)). Derived from the odd part of the bisection of the o.g.f. for the C(n,x)^2 polynomials. Note that p(n,x) = (tau(n,sqrt(x)))^2.

LINKS

Table of n, a(n) for n=0..48.

FORMULA

a(n,m) = [x^m] (p(n,x)), n>=0, 0 <= m <= 2*n, with p(n,x) = (C(2*n+1,sqrt(x))/sqrt(x))^2 = (tau(n,sqrt(x)))^2, For Chebyshev's C and tau polynomials see a comment above.

For n >= 0, 0 <= m <= 2*n, a(n,m) = [x^m*z^n] G2(x,z), where the o.g.f. G2(x,z) given in a comment above.

a(n,m) = (-1)^m * A156308(2*n+1,m+1). - Max Alekseyev, Mar 06 2018

EXAMPLE

The array begins:

n\m  0     1    2     3     4     5    6     7    8   9  10

0:   1

1:   9    -6    1

2:  25   -50   35   -10     1

3:  49  -196  294  -210    77   -14    1

4:  81  -540 1386 -1782  1287  -546  135   -18    1

5: 121 -1210 4719 -9438 11011 -8008 3740 -1122  209 -22   1

...

Row polynomial for n=1: p(1,x) = (tau(1,sqrt(x)))^2 = (3-1*x)^2 = 9 - 6*x +1*x^2.

Row polynomial for n=2: p(2,x) = (tau(2,sqrt(x)))^2 = (5 - 5*x + 1*x^2)^2 = 25 - 50*x + 35*x^2 - 10*x^3 + 1*x^4.

CROSSREFS

Odd rows of A156308 with alternating signs of elements.

Cf. A005408, A111125.

Sequence in context: A021840 A198582 A199082 * A064230 A286331 A089479

Adjacent sequences:  A220666 A220667 A220668 * A220670 A220671 A220672

KEYWORD

sign,easy,tabf

AUTHOR

Wolfdieter Lang, Jan 04 2013

STATUS

approved

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Last modified November 14 03:52 EST 2018. Contains 317159 sequences. (Running on oeis4.)