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%I #41 Mar 29 2024 08:47:19
%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)
%F 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
%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-François 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. A157142, A000330(n), A005585, A050486, A054333.
%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) by _Wolfdieter Lang_, Aug 02 2014
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