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A039991 Triangle of coefficients of cos(x)^n in polynomial for cos(nx). 18
1, 1, 0, 2, 0, -1, 4, 0, -3, 0, 8, 0, -8, 0, 1, 16, 0, -20, 0, 5, 0, 32, 0, -48, 0, 18, 0, -1, 64, 0, -112, 0, 56, 0, -7, 0, 128, 0, -256, 0, 160, 0, -32, 0, 1, 256, 0, -576, 0, 432, 0, -120, 0, 9, 0, 512, 0, -1280, 0, 1120, 0, -400, 0, 50, 0, -1, 1024, 0, -2816, 0, 2816, 0, -1232, 0, 220 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Also triangle of coefficients of Chebyshev polynomials of first kind (T(n,x)) in decreasing order of powers of x. A053120 gives the coefficients in increasing order.

The polynomials R(n,x) := sum(a(n,m)*sqrt(x)^m, m=0..n) have g.f. (1-z)/(1 - 2*z + x*z^2) = ((1-z)/(1-2*z))/(1 - x*(-z^2/(1-2*z))) (from the row reversion of the g.f. of A053120 and x^2 -> x). Therefore this triangle becomes the Riordan triangle ((1-z)/(1-2*z), -z^2/(1-2*z)) if the vanishing columns are deleted (see A028297) and zeros are appended in each row numbered n>=1 in order to obtain a triangle. This is then A201701 with negative odd numbered columns. - Wolfdieter Lang, Aug 06 2014

REFERENCES

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 795.

Martin Aigner and Gunter M. Ziegler, Proofs From the Book, Springer 2004. See Chapter 18, Appendix.

E. A. Guilleman, Synthesis of Passive Networks, Wiley, 1957, p. 593.

Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

LINKS

T. D. Noe, Table of n, a(n) for n = 0..5150

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Index entries for sequences related to Chebyshev polynomials.

FORMULA

a(n, m) = 0 if n<m or m odd, (-1)^{m/2} if m=n is even, ((-1)^(3*m/2))*(2^(n-m-1))*n*binomial(n-1-m/2, n-1-m)/(n-m) else. a(n, m) = 2*a(n-1, m)-a(n-2, m-2), n >= 2, m >= 0; a(n, -2) := 0=: a(n, -1), a(0, 0)=1=a(1, 0).

G.f. for m-th column: 0 if m odd, (1-x)/(1-2*x) if m=0, else ((-1)^(m/2))*(x^m)*(1-x)/(1-2*x)^(m/2+1). For g.f. for row polynomials and row sums, see A053120.

G.f. row polynomials: (1-z)/(1 - 2*z + (x*z)^2. - Wolfdieter Lang, Aug 06 2014

Recurrence for the row polynomials Trev(n, x):= x^n*T(n, 1/x) = sum(a(n,m)*x^m, m=0..n): Trev(n, x) = 2*Trev(n-1, x) - x^2*Trev(n-2, x), n >= 1, Trev(-1, x) = 1/x^2 and Trev(0, x) = 1. From the T(n, x) recurrence. Compare this with A081265. - Wolfdieter Lang, Aug 07 2014

EXAMPLE

Letting c = cos x, we have: cos 0x = 1, cos 1x = 1c; cos 2x = 2c^2-1; cos 3x = 4c^3-3c, cos 4x = 8c^4-8c^2+1, etc.

From Wolfdieter Lang, Aug 06 2014: (Start)

The triangle a(n,m) begins:

n\m     0 1      2 3     4 5      6 7     8  9    10 11  12 13  14 15 ...

0:      1

1:      1 0

2:      2 0     -1

3:      4 0     -3 0

4:      8 0     -8 0     1

5:     16 0    -20 0     5 0

6:     32 0    -48 0    18 0     -1

7:     64 0   -112 0    56 0     -7 0

8:    128 0   -256 0   160 0    -32 0     1

9:    256 0   -576 0   432 0   -120 0     9  0

10:   512 0  -1280 0  1120 0   -400 0    50  0    -1

11:  1024 0  -2816 0  2816 0  -1232 0   220  0   -11  0

12:  2048 0  -6144 0  6912 0  -3584 0   840  0   -72  0   1

13:  4096 0 -13312 0 16640 0  -9984 0  2912  0  -364  0  13  0

14:  8192 0 -28672 0 39424 0 -26880 0  9408  0 -1568  0  98  0  -1

15: 16384 0 -61440 0 92160 0 -70400 0 28800  0 -6048  0 560  0 -15  0

...

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

Chebyshev T-polynomials (decreasing even or odd powers):

n=3: T(3, n) = 4*x^3  - 3*x^1; n=4: T(4, x) = 8*x^4 - 8*x^2 + 1. (End)

MAPLE

seq(seq(coeff(orthopoly[T](i, x), x, i-j), j=0..i), i=0..20); # Robert Israel, Aug 07 2014

MATHEMATICA

row[n_] := CoefficientList[ ChebyshevT[n, x], x] // Reverse; Table[row[n], {n, 0, 11}] // Flatten(* Jean-Fran├žois Alcover, Sep 14 2012 *)

CROSSREFS

Cf. A028297 (without vanishing columns). A008310 (zero columns deleted then rows reversed).

Triangle without zeros: A028297. Without signs: A081265.

Cf. A053120 (increasing powers of x).

Sequence in context: A181670 A261251 * A081265 A273821 A108643 A133838

Adjacent sequences:  A039988 A039989 A039990 * A039992 A039993 A039994

KEYWORD

tabl,easy,sign,nice

AUTHOR

David W. Wilson

EXTENSIONS

Entry improved by comments from Wolfdieter Lang, Jan 11 2000.

Edited: A053120 added in comment and crossrefs. Cfs. A028297 and A008310 specified. - Wolfdieter Lang, Aug 06 2014

STATUS

approved

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Last modified November 14 21:12 EST 2018. Contains 317217 sequences. (Running on oeis4.)