A084930 Triangle of coefficients of Chebyshev polynomials T_{2n+1} (x).

1, -3, 4, 5, -20, 16, -7, 56, -112, 64, 9, -120, 432, -576, 256, -11, 220, -1232, 2816, -2816, 1024, 13, -364, 2912, -9984, 16640, -13312, 4096, -15, 560, -6048, 28800, -70400, 92160, -61440, 16384, 17, -816, 11424, -71808, 239360, -452608, 487424, -278528, 65536, -19, 1140, -20064, 160512, -695552

Offset

0,2

Comments

From Herb Conn, Jan 28 2005: (Start)

"Letting x = 2 Cos 2A, we have the following trigonometric identities:

"Sin 3A = 3*Sin A - 4*Sin^3 A

"Sin 5A = 5*Sin A - 20*Sin^3 A + 16*Sin^5 A

"Sin 7A = 7*Sin A - 56*Sin^3 A + 112*Sin^5 A - 64*Sin^7 A

"Sin 9A = 9*Sin A - 120*Sin^3 A + 432*Sin^5 A - 576*Sin^7 A + 256*Sin^9 A, etc." (End)

Cayley (1876) states "Write sin u = x, then we have sin u = x, [...] sin 3u = 3x - 4x^3, [...] sin 5u = 5x - 20x^3 + 16 x^5, [...]". Since T_n(cos(u)) = cos(nu) for all integer n, sin(u) = cos(u - Pi/2), and sin(u + k*Pi) = (-1)^k sin(u) it follows that T_n(sin(u)) = (-1)^((n-1)/2) sin(nu) for all odd integer n. - Michael Somos, Jun 19 2012

From Wolfdieter Lang, Aug 05 2014: (Start)

The coefficient triangle t(n,k) for the row polynomials Todd(n, x) := T_{2*n+1}(sqrt(x))/sqrt(x) = sum(t(n,k)*x^k, k=0..n) is the Riordan triangle ((1-z)/(1+z)^2, 4*z/(1+z)^2) (rewrite the g.f. for the present triangle a(n,k) given in the formula section). The triangle entries t(n,k) = a(n,k), but the interpretation of the row polynomials is different for both cases.

From the relation Todd(n, x) = S(n, 2*(2*x-1)) - S(n-1, 2*(2*x-1)) with the Chebyshev S-polynomials (see A049310 and the formula section of A130777) follows the recurrence: Todd(n, x) = 2*(-1)^n*(1-x)*Todd(n-1, 1-x) + (2x-1)*Todd(n-1, x), n >= 1, Todd(0, x) = 1.

This corresponds to the triangle recurrence t(n,k) = (2*(k+1)*(-1)^(n-k) - 1)*t(n-1,k) + 2*(1 +(-1)^(n-k))*t(n-1,k-1) + 2*(-1)^(n-k)*sum(binomial(l+1,k)*t(n-1,l), l=k+1..n-1), n >= k >= 1, t(n,k) = 0 if n < k, t(n,0) = (-1)^n*(2*n+1). Compare this with the shorter recurrence involving the rational A-sequence for this Riordan triangle which has g.f. x^2/(2-x-2*sqrt(1-x)). t(n,k) = sum(A(j)*t(n-1,k-1+j), j=0..n-k), n >= k >= 1. The Z-sequence has g.f. -(1 + 2/sqrt(1-x)). For the A- and Z-sequence see a link under A006232. (End)

References

A. Cayley, On an Expression for 1 +- sin(2p+1)u in Terms of sin u, Messenger of Mathematics, 5 (1876), pp. 7-8 = Mathematical Papers Vol. 10, n. 630, pp. 1-2.

Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2nd ed., Wiley, New York, 1990. p. 37, eq. (1.96) and p. 4, eq. (1.10).

Links

Table of n, a(n) for n=0..49.

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].

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 795.

Index entries for sequences related to Chebyshev polynomials.

Formula

Alternate rows of A008310.

a(n,k)=((-1)^(n-k))*(2^(2*k))*binomial(n+1+k,2*k+1)*(2*n+1)/(n+1+k) if n>=k>=0 else 0.

From Wolfdieter Lang, Aug 02 2014: (Start)

a(n,k) = [x^(2*k+1)] T_{2*n+1}(x), n >= k >= 0.

G.f. for row polynomials: x*(1-z)/(1 + 2*(1- 2*x^2)*z + z^2). (End)

The first column sequences are: A157142, 4*(-1)^(n+1)*A000330(n), 16*(-1)^n*A005585(n-1), 64*(-1)^(n+1)*A050486(n-3), 256*(-1)^n*A054333(n-4), ... - Wolfdieter Lang, Aug 05 2014

Example

The triangle a(n,k):
n   2n+1\k 0     1      2       3       4        5        6         7        8        9      10 ...
0    1:    1
1    3:   -3     4
2    5:    5   -20     16
3    7:   -7    56   -112      64
4    9:    9  -120    432    -576     256
5   11:  -11   220  -1232    2816   -2816     1024
6   13:   13  -364   2912   -9984   16640   -13312     4096
7   15:  -15   560  -6048   28800  -70400    92160   -61440     16384
8   17:   17  -816  11424  -71808  239360  -452608   487424   -278528    65536
9   19:  -19  1140 -20064  160512 -695552  1770496 -2723840   2490368 -1245184   262144
10  21:   21 -1540  33264 -329472 1793792 -5870592 12042240 -15597568 12386304 -5505024 1048576
... formatted and extended by Wolfdieter Lang, Aug 02 2014.
---------------------------------------------------------------------------------------------------
First few polynomials T_{2n+1}(x) are
1*x - 3*x + 4*x^3
5*x - 20*x^3 + 16*x^5
- 7*x + 56*x^3 - 112*x^5 + 64*x^7
9*x - 120*x^3 + 432*x^5 - 576*x^7  + 256*x^9

Mathematica

row[n_] := (T = ChebyshevT[2*n+1, x]; Coefficient[T, x, #]& /@ Range[1, Exponent[T, x], 2]); Table[row[n], {n, 0, 9} ] // Flatten (* Jean-François Alcover, Aug 06 2014 *)

Crossrefs

Cf. A002315, A028297.

Cf. A053120 (coefficient triangle of T-polynomials), A127674 (even-indexed T polynomials).

Cf. A127675 (row reversed triangle with different signs).

Cf. A006232, A008310, A049310, A130777.

Cf. A157142, A000330(n), A005585, A050486, A054333.

Sequence in context: A161961 A161474 A247573 * A114336 A048086 A048005

Adjacent sequences: A084927 A084928 A084929 * A084931 A084932 A084933

Keyword

sign,tabl,easy

Author

Gary W. Adamson, Jun 12 2003

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jun 26 2003

Edited; example rewritten (to conform with the triangle) by Wolfdieter Lang, Aug 02 2014

Status

approved