A181878 - OEIS

A181878

Coefficient array for square of Chebyshev S-polynomials.

1, 1, 1, -2, 1, 4, -4, 1, 1, -6, 11, -6, 1, 9, -24, 22, -8, 1, 1, -12, 46, -62, 37, -10, 1, 16, -80, 148, -128, 56, -12, 1, 1, -20, 130, -314, 367, -230, 79, -14, 1, 25, -200, 610, -920, 771, -376, 106, -16, 1, 1, -30, 295, -1106, 2083, -2232, 1444, -574, 137, -18, 1, 36, -420, 1897, -4352, 5776, -4744, 2486, -832, 172, -20, 1, 1, -42, 581, -3108, 8518, -13672, 13820, -9142, 4013, -1158, 211, -22, 1

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OFFSET

0,4

COMMENTS

For the coefficients of Chebyshev polynomials S(n,x) see A049310.

The row length sequence for this array is A109613 = {1,1,3,3,5,5,...}.

The row polynomials (in x^2) for even row numbers are

S(2*k,x)^2 = Sum_{m=0..2*k} a(2*k,m)*x^(2*m), k >= 0.

For odd row numbers the row polynomials (in x^2) are

(S(2*k+1,x)^2)/x^2 = Sum_{m=0..2*k} a(2*k+1,m)*x^(2*m), k >= 0.

The o.g.f. for the polynomials S(n,x)^2 is

S(x,z):=((1+z)/(1-z))/(1 + (2-x^2)z +z^2). See the link for a proof. Therefore the coefficients constitute the Riordan array (1/(1-x^2),x/(1+x)^2) found as A158454.

The o.g.f. for (S(2*k,sqrt(x))^2 is

(1-2(1-x)z+z^2)/((1-z)*(1 - (2-4x+x^2)z + z^2)).

The o.g.f. for ((S(2*k+1,sqrt(x))^2)/x is

((1+z)/(1-z))/(1 - (2-4x+x^2)z + z^2).

The row sums A011655(n+1) are the same as those for the triangle A158454.

The alternating row sums for even numbered rows (-1)^n*A007598(n+1) coincide with those of triangle A158454. For odd row numbers n=2k+1 these sums are A049684(k+1), k >= 0 (squares of even-indexed Fibonacci numbers).

LINKS

Table of n, a(n) for n=0..84.

Wolfdieter Lang, First ten rows with more details and proofs.

FORMULA

a(2*k,m) = (-1)^m*Sum_{j=0..k} binomial(2*k+m-1-2*j, 2*m-1), k >= 0.

a(2*k+1,m) = (-1)^m*Sum_{j=0..k} binomial(2*k+1+m-2*j, 2*m+1), k >= 0.

This derives from the formula for the entries of the Riordan array A158454.

For the o.g.f.s see the comment.

EXAMPLE

The irregular triangle a(n,m) begins:

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

0: 1