login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A130777 Coefficients of first difference of Chebyshev S polynomials. 15
1, -1, 1, -1, -1, 1, 1, -2, -1, 1, 1, 2, -3, -1, 1, -1, 3, 3, -4, -1, 1, -1, -3, 6, 4, -5, -1, 1, 1, -4, -6, 10, 5, -6, -1, 1, 1, 4, -10, -10, 15, 6, -7, -1, 1, -1, 5, 10, -20, -15, 21, 7, -8, -1, 1, -1, -5, 15, 20, -35, -21, 28, 8, -9, -1, 1, 1, -6, -15, 35, 35, -56, -28, 36, 9, -10, -1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

Inverse of triangle in A061554.

Signed version of A046854.

From Paul Barry, May 21 2009: (Start)

Riordan array ((1-x)/(1+x^2),x/(1+x^2)).

This triangle is the coefficient triangle for the Hankel transforms of the family of generalized Catalan numbers that satisfy a(n;r)=r*a(n-1;r)+sum{k=1..n-2, a(k)*a(n-1-k;r)}, a(0;r)=a(1;r)=1. The Hankel transform of a(n;r) is h(n)=sum{k=0..n, T(n,k)*r^k} with g.f. (1-x)/(1-r*x+x^2). These sequences include A086246, A000108, A002212. (End)

From Wolfdieter Lang, Jun 11 2011: (Start)

The Riordan array ((1+x)/(1+x^2),x/(1+x^2)) with entries Phat(n,k)= ((-1)^(n-k))*T(n,k) and o.g.f. Phat(x,z)=(1+z)/(1-x*z+z^2) for the row polynomials Phat(n,x) is related to Chebyshev C and S polynomials as follows.

Phat(n,x) = (R(n+1,x)-R(n,x))/(x+2) = S(2*n,sqrt(2+x))

  with R(n,x)=C_n(x) in the Abramowitz and Stegun notation, p. 778, 22.5.11. See A049310 for the S polynomials. Proof from the o.g.f.s.

Recurrence for the row polynomials Phat(n,x):

  Phat(n,x) = x*Phat(n-1,x) - Phat(n-2,x) for n>=1; Phat(-1,x)=-1, Phat(0,x)=1.

The A-sequence for this Riordan array Phat (see the W. Lang link under A006232 for A- and Z-sequences for Riordan matrices) is given by 1, 0, -1, 0, -1, 0, -2, 0, -5,.., starting with 1 and interlacing the negated A000108 with zeros (o.g.f. 1/c(x^2) = 1-c(x^2)*x^2, with the o.g.f. c(x) of A000108).

The Z-sequence has o.g.f. sqrt((1-2*x)/(1+2*x)), and it is given by A063886(n)*(-1)^n.

The A-sequence of the Riordan array T(n,k) is identical with the one for the Riordan array Phat, and the Z-sequence is -A063886(n).

(End)

The row polynomials P(n,x) are the characteristic polynomials of the adjacency matrices of the graphs which look like P_n (n vertices (nodes), n-1 lines (edges)), but vertex no. 1 has a loop. - Wolfdieter Lang, Nov 17 2011

From Wolfdieter Lang, Dec 14 2013: (Start)

The zeros of P(n,x) are x(n,j) = -2*cos(2*Pi*j/(2*n+1)), j=1..n. From P(n,x) = (-1)^n*S(2*n,sqrt(2-x)) (see, e.g., the Lemma 6 of the W. Lang link).

The discriminants of the P-polynomials are given in A052750. (End)

REFERENCES

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964. Tenth printing, Wiley, 2002 (also electronically available).

LINKS

Table of n, a(n) for n=0..77.

Wolfdieter Lang, The field Q(2cos(pi/n)), its Galois group and length ratios in the regular n-gon, arXiv:1210.1018 [math.GR], see Definition 1, Lemma 6 and Remark 4.

P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), p. 22-31 (formula 5).

Index entries for sequences related to Chebyshev polynomials.

FORMULA

Number triangle T(n,k) = (-1)^C(n-k+1,2)*C(floor((n+k)/2),k). - Paul Barry, May 21 2009

From Wolfdieter Lang, Jun 11 2011: (Start)

Row polynomials: P(n,x) = sum(k=0..n, T(n,k)*x^k) = R(2*n+1,sqrt(2+x)) / sqrt(2+x), with Chebyshev polynomials R with coefficients given in A127672 (scaled T-polynomials).

R(n,x) is called C_n(x) in Abramowitz and Stegun's handbook, p. 778, 22.5.11.

P(n,x) = S(n,x)-S(n-1,x), n>=0, S(-1,x)=0, with the Chebyshev S-polynomials (see the coefficient triangle A049310).

O.g.f. for row polynomials: P(x,z):= sum(n>=0, P(n,x)*z^n ) = (1-z)/(1-x*z+z^2).

  (from the o.g.f. for R(2*n+1,x), n>=0, computed from the o.g.f. for the R-polynomials (2-x*z)/(1-x*z+z^2) (see A127672))

Proof of the Chebyshev connection from the o.g.f. for Riordan array property of this triangle (see the P. Barry comment above).

For the A- and Z-sequences of this Riordan array see a comment above. (End)

abs(T(n,k)) = A046854(n,k) = abs(A066170(n,k)) T(n,n-k) = A108299(n,k); abs(T(n,n-k)) = A065941(n,k). - Johannes W. Meijer, Aug 08 2011

From Wolfdieter Lang, Jul 31 2014: (Start)

Similar to the triangles A157751, A244419 and A180070 one can give for the row polynomials P(n,x) besides the usual three term recurrence another one needing only one recurrence step. This uses also a negative argument, namely P(n,x) = (-1)^(n-1)*(-1 + x/2)*P(n-1,-x) + (x/2)*P(n-1,x), n >= 1, P(0,x) = 1. Proof by computing the o.g.f. and comparing with the known one. This entails the alternative triangle recurrence T(n,k) = (-1)^(n-k)*T(n-1,k) + (1/2)*(1 + (-1)^(n-k))*T(n-1,k-1), n >= m >= 1, T(n,k) = 0 if n < k and T(n,0) = (-1)^floor((n+1)/2) = A057077(n+1). [P(n,x) recurrence corrected Aug 03 2014]

(End)

EXAMPLE

The triangle T(n,k) begins:

n\k  0   1   1   3    4    5    6    7    8    9  10  11  12  13 14 15 ...

0:   1

1:  -1   1

2:  -1  -1   1

3:   1  -2  -1   1

4:   1   2  -3  -1    1

5:  -1   3   3  -4   -1    1

6:  -1  -3   6   4   -5   -1    1

7:   1  -4  -6  10    5   -6   -1    1

8:   1   4 -10 -10   15    6   -7   -1    1

9:  -1   5  10 -20  -15   21    7   -8   -1    1

10: -1  -5  15  20  -35  -21   28    8   -9   -1   1

11:  1  -6 -15  35   35  -56  -28   36    9  -10  -1   1

12:  1   6 -21 -35   70   56  -84  -36   45   10 -11  -1   1

13: -1   7  21 -56  -70  126   84 -120  -45   55  11 -12  -1   1

14: -1  -7  28  56 -126 -126  210  120 -165  -55  66  12 -13  -1  1

15:  1  -8 -28  84  126 -252 -210  330  165 -220 -66  78  13 -14 -1  1

...  reformatted and extended - Wolfdieter Lang, Jul 31 2014.

---------------------------------------------------------------------------

From Paul Barry, May 21 2009: (Start)

Production matrix is

-1, 1,

-2, 0, 1,

-2, -1, 0, 1,

-4, 0, -1, 0, 1,

-6, -1, 0, -1, 0, 1,

-12, 0, -1, 0, -1, 0, 1,

-20, -2, 0, -1, 0, -1, 0, 1,

-40, 0, -2, 0, -1, 0, -1, 0, 1,

-70, -5, 0, -2, 0, -1, 0, -1, 0, 1 (End)

Row polynomials as first difference of S polynomials:

P(3,x) = S(3,x) - S(2,x) = (x^3 - 2*x) - (x^2 -1) = 1 - 2*x - x^2 +x^3.

Alternative triangle recurrence (see a comment above): T(6,2) = T(5,2) + T(5,1) = 3 + 3 = 6. T(6,3) = -T(5,3) + 0*T(5,1) = -(-4) = 4. - Wolfdieter Lang, Jul 31 2014

MAPLE

A130777 := proc(n, k): (-1)^binomial(n-k+1, 2)*binomial(floor((n+k)/2), k) end: seq(seq(A130777(n, k), k=0..n), n=0..11); # Johannes W. Meijer, Aug 08 2011

MATHEMATICA

T[n_, k_] := (-1)^Binomial[n - k + 1, 2]*Binomial[Floor[(n + k)/2], k];

Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 14 2017, from Maple *)

PROG

(Sage)

@CachedFunction

def A130777(n, k):

    if n< 0: return 0

    if n==0: return 1 if k == 0 else 0

    h = A130777(n-1, k) if n==1 else 0

    return A130777(n-1, k-1) - A130777(n-2, k) - h

for n in (0..9): [A130777(n, k) for k in (0..n)] # Peter Luschny, Nov 20 2012

CROSSREFS

Cf. A066170, A046854, A057077 (first column).

Row sums: A010892(n+1); repeat(1,0,-1,-1,0,1). Alternating row sums: A061347(n+2); repeat(1,-2,1).

Sequence in context: A225631 A306209 A267482 * A187660 A046854 A066170

Adjacent sequences:  A130774 A130775 A130776 * A130778 A130779 A130780

KEYWORD

sign,tabl,easy

AUTHOR

Philippe Deléham, Jul 14 2007

EXTENSIONS

New name and Chebyshev comments by Wolfdieter Lang, Jun 11 2010

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified February 15 22:28 EST 2019. Contains 320138 sequences. (Running on oeis4.)