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A128494 Coefficient table for sums of Chebyshev's S-Polynomials. 4
1, 1, 1, 0, 1, 1, 0, -1, 1, 1, 1, -1, -2, 1, 1, 1, 2, -2, -3, 1, 1, 0, 2, 4, -3, -4, 1, 1, 0, -2, 4, 7, -4, -5, 1, 1, 1, -2, -6, 7, 11, -5, -6, 1, 1, 1, 3, -6, -13, 11, 16, -6, -7, 1, 1, 0, 3, 9, -13, -24, 16, 22, -7, -8, 1, 1, 0, -3, 9, 22, -24, -40, 22, 29, -8, -9, 1, 1, 1, -3, -12, 22, 46, -40, -62, 29, 37, -9, -10, 1, 1, 1, 4, -12 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,13

COMMENTS

See A049310 for the coefficient table of Chebyshev's S(n,x)=U(n,x/2) polynomials.

This is a 'repetition triangle' based on a signed version of triangle A059260: a(2*p,2*k)=a(2*p+1,2*k)=A059260(p+k,2*k)*(-1)^(p+k) and a(2*p+1,2*k+1)=a(2*p+2,2*k+1)=A059260(p+k+1,2*k+1)*(-1)^(p+k), k>=0.

LINKS

Table of n, a(n) for n=0..93.

Wolfdieter Lang, First 15 rows.

Per Alexandersson, Luis Angel González-Serrano, Egor A. Maximenko, Mario Alberto Moctezuma-Salazar, Symmetric polynomials in the symplectic alphabet and their expression via Dickson-Zhukovsky variables, arXiv:1912.12725 [math.CO], 2019.

FORMULA

S(1;n,x) = sum(S(k,x),k=0..n) = sum(a(n,m)*x^m,m=0..n), n>=0.

a(n,m) = [x^m](S(n,y)*S(n+1,y)/y) with y:=sqrt(2+x).

G.f. for column sequence nr. m: (x^m)/((1-x)*(1+x^2)^(m+1)), which shows that this is a lower diagonal matrix of the Riordan type, named (1/((1+x^2)*(1-x)), x/(1+x^2)).

From Wolfdieter Lang, Oct 16 2012: (Start)

a(n,m) = [x^m](1- (S(n+1,x) - S(n,x)))/(2-x). From the Binet  -de Moivre formula for S and use of the geometric sum.

a(n,m) = [x^m](1- R(2*n+3,sqrt(2+x))/sqrt(2+x))/(2-x) with the monic integer T-polynomials R with coefficient triangle given in A127672. From the odd part of the bisection of the T-polynomials. (End)

EXAMPLE

The triangle a(n,m) begins:

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

0:  1

1:  1   1

2:  0   1  1

3:  0  -1  1   1

4:  1  -1 -2   1   1

5:  1   2 -2  -3   1   1

6:  0   2  4  -3  -4   1   1

7:  0  -2  4   7  -4  -5   1   1

8:  1  -2 -6   7  11  -5  -6   1   1

9:  1   3 -6 -13  11  16  -6  -7   1  1

10: 0   3  9 -13 -24  16  22  -7  -8  1  1

... reformatted by Wolfdieter Lang, Oct 16 2012

Row polynomial S(1;4,x)=1-x-2*x^2+x^3+x^4 = sum(S(k,x),k=0..4).

S(4,y)*S(5,y)/y=3-13*y^2+16*y^4-7*y^6+y^8, with y=sqrt(2+x) this becomes S(1;4,x).

From Wolfdieter Lang, Oct 16 2012: (Start)

S(1;4,x) = (1- (S(5,x) - S(4,x)))/(2-x) = (1-x)*(2-x)*(1+x)*(1-x-x^2)/(2-x) = (1-x)*(1+x)*(1-x-x^2).

S(5,x) - S(4,x) = R(11,sqrt(2+x))/sqrt(2+x) = -1+3*x+3*x^2-4*x^3-x^4+x^5. (End)

CROSSREFS

Row sums (signed): A021823(n+2). Row sums (unsigned): A070550(n).

Cf. A128495 for S(2; n, x) coefficient table.

The column sequences (unsigned) are, for m=0..4: A021923, A002265, A008642, A128498, A128499.

For m>=1 the column sequences (without leading zeros) are of the form a(m, 2*k)=a(m, 2*k+1)=((-1)^k)*b(m, k) with the sequences b(m, k), given for m=1..11 by A008619, A002620, A002623, A001752, A001753, A001769, A001779, A001780, A001781, A001786, A001808.

Sequence in context: A228431 A328702 A165620 * A257696 A110730 A198339

Adjacent sequences:  A128491 A128492 A128493 * A128495 A128496 A128497

KEYWORD

sign,tabl,easy

AUTHOR

Wolfdieter Lang Apr 04 2007

STATUS

approved

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Last modified July 12 17:40 EDT 2020. Contains 335665 sequences. (Running on oeis4.)