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A084930 Triangle of coefficients of Chebyshev polynomials T_{2n+1} (x). 9

%I

%S 1,-3,4,5,-20,16,-7,56,-112,64,9,-120,432,-576,256,-11,220,-1232,2816,

%T -2816,1024,13,-364,2912,-9984,16640,-13312,4096,-15,560,-6048,28800,

%U -70400,92160,-61440,16384,17,-816,11424,-71808,239360,-452608,487424,-278528,65536,-19,1140,-20064,160512,-695552

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

%C From _Herb Conn_, Jan 28 2005: (Start)

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

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

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

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

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

%C 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

%C From _Wolfdieter Lang_, Aug 05 2014: (Start)

%C 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.

%C 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.

%C 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)

%D 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.

%D 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).

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 795.

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%F Alternate rows of A008310.

%F 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.

%F From _Wolfdieter Lang_, Aug 02 2014: (Start)

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

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

%e The triangle a(n,k):

%e n 2n+1\k 0 1 2 3 4 5 6 7 8 9 10 ...

%e 0 1: 1

%e 1 3: -3 4

%e 2 5: 5 -20 16

%e 3 7: -7 56 -112 64

%e 4 9: 9 -120 432 -576 256

%e 5 11: -11 220 -1232 2816 -2816 1024

%e 6 13: 13 -364 2912 -9984 16640 -13312 4096

%e 7 15: -15 560 -6048 28800 -70400 92160 -61440 16384

%e 8 17: 17 -816 11424 -71808 239360 -452608 487424 -278528 65536

%e 9 19: -19 1140 -20064 160512 -695552 1770496 -2723840 2490368 -1245184 262144

%e 10 21: 21 -1540 33264 -329472 1793792 -5870592 12042240 -15597568 12386304 -5505024 1048576

%e ... formatted and extended by _Wolfdieter Lang_, Aug 02 2014.

%e ---------------------------------------------------------------------------------------------------

%e First few polynomials T_{2n+1}(x) are

%e 1*x - 3*x + 4*x^3

%e 5*x - 20*x^3 + 16*x^5

%e - 7*x + 56*x^3 - 112*x^5 + 64*x^7

%e 9*x - 120*x^3 + 432*x^5 - 576*x^7 + 256*x^9

%t 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-Fran├žois Alcover_, Aug 06 2014 *)

%Y Cf. A002315, A028297.

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

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

%Y Cf. A006232, A008310, A049310, A130777.

%Y Cf. 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.

%K sign,tabl,easy

%O 0,2

%A _Gary W. Adamson_, Jun 12 2003

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

%E Edited. Example rewritten to conform with the triangle. _Wolfdieter Lang_, Aug 02 2014

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Last modified July 4 11:26 EDT 2020. Contains 335448 sequences. (Running on oeis4.)