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A084930 Triangle of coefficients of Chebyshev polynomials T_{2n+1} (x). 9
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 (list; table; graph; refs; listen; history; text; internal format)
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
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.
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. A053120 (coefficient triangle of T-polynomials), A127674 (even-indexed T polynomials).
Cf. A127675 (row reversed triangle with different signs).
Sequence in context: A161961 A161474 A247573 * A114336 A048086 A048005
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

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