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A127677 Scaled coefficient table for Chebyshev polynomials 2*T(2*n, sqrt(x)/2) (increasing even scaled powers, without zero entries).

%I A127677 #106 Feb 16 2025 08:33:04
%S A127677 2,-2,1,2,-4,1,-2,9,-6,1,2,-16,20,-8,1,-2,25,-50,35,-10,1,2,-36,105,
%T A127677 -112,54,-12,1,-2,49,-196,294,-210,77,-14,1,2,-64,336,-672,660,-352,
%U A127677 104,-16,1,-2,81,-540,1386,-1782,1287,-546,135,-18,1,2,-100,825,-2640,4290,-4004,2275,-800,170,-20,1
%N A127677 Scaled coefficient table for Chebyshev polynomials 2*T(2*n, sqrt(x)/2) (increasing even scaled powers, without zero entries).
%C A127677 2*T(2*n,x) = Sum_{m=0..n} a(n,m)*(2*x)^(2*m).
%C A127677 Closely related to A284982, which has opposite signs and rows begin with 0 of alternating signs instead of +/2. - _Eric W. Weisstein_, Apr 07 2017
%C A127677 Bisection triangle of A127672 (without zero entries, even part). The odd part is ((-1)^(n-m))*A111125(n,m).
%C A127677 If the leading 2 is replaced by a 1 we get the essentially identical sequence A110162. - _N. J. A. Sloane_, Jun 09 2007
%C A127677 Also row n gives coefficients of characteristic polynomial of the Cartan matrix for the root system B_n (or, equally, C_n). - _Roger L. Bagula_, May 23 2007
%C A127677 From _Wolfdieter Lang_, Oct 04 2013: (Start)
%C A127677 This triangle a(n,m) is used to express the length ratio side/R given by s(4*n+2) = 2*sin(Pi/(4*n+2)) = 2*cos(2*n*Pi/(4*n+2)) in a regular (4*n+2)-gon, inscribed in a circle with radius R, in terms of rho(4*n+2) = 2*cos(Pi/4*n+2), the length ratio of (the smallest diagonal)/side (for n=2 there is no such diagonal).
%C A127677 s(4*n+2) = Sum_{m=0..n}a(n,m)*rho(4*n+2)^(2*m). This formula is needed to show that the total sum of all length ratios in a (4*n+2)-gon is an integer in the algebraic number field Q(rho(4*n+2)). Note that rho(4*n+2) has degree delta(4*n+2) = A055034(4*n+2). Therefore one has to take s(4*n+2) modulo C(4*n+2, x=rho(4*n+2)), the minimal polynomial of rho(4*n+2) (see A187360). Thanks go to Seppo Mustonen for asking me to look into this problem. See ((-1)^(n-m))*A111125(n,m) for the (4*n)-gon situation. (End)
%D A127677 R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 62
%D A127677 Sigurdur Helgasson,Differential Geometry, Lie Groups and Symmetric Spaces, Graduate Studies in Mathematics, volume 34. A. M. S. :ISBN 0-8218-2848-7, 1978,p. 463.
%H A127677 Wolfdieter Lang, <a href="/A127677/a127677.pdf">First 10 rows and more.</a>
%H A127677 P. Damianou , <a href="http://arxiv.org/abs/1110.6620">On the characteristic polynomials of Cartan matrices and Chebyshev polynomials</a>, arXiv preprint arXiv:1110.6620 [math.RT], 2014.
%H A127677 P. Damianou and C. Evripidou, <a href="http://arxiv.org/abs/1409.3956">Characteristic and Coxeter polynomials for affine Lie algebras</a>, arXiv preprint arXiv:1409.3956 [math.RT], 2014.
%H A127677 Yidong Sun, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/43-4/paper43-4-10b.pdf">Numerical triangles and several classical sequences</a>, Fib. Quart., Nov. 2005, pp. 359-370. See Table 1.6 (an unsigned version).
%H A127677 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CartanMatrix.html">Cartan Matrix</a>
%H A127677 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DynkinDiagram.html">Dynkin Diagram</a>
%F A127677 a(n,m) = 0 if n < m; a(n,0) = 2*(-1)^n; a(n,m) = ((-1)^(n+m))*n*binomial(n+m-1, 2*m-1)/m.
%F A127677 a(n,m) = 0 if n < m, a(0,0) = 2, a(n,m) = (-1)^(n-m)*(2*n/(n+m))*binomial(n+m, n-m), n >= 1. From Waring's formula applied to Chebyshev's T-polynomials. See also A110162. - _Wolfdieter Lang_, Nov 21 2012
%F A127677 The o.g.f. Sum_{n>=0} p(n,x)*z^n, n>=0, for the row polynomials p(n,x) := Sum_{m=0..n} a(n,m)*x^m is (2 + z*(2-x))/((z+1)^2 - z*x). Here p(n,x) = R(2*n,sqrt(x)) := 2*T(2*n,sqrt(x)/2) with Chebyshev's T-polynomials. For the R-polynomials see A127672. - _Wolfdieter Lang_, Nov 28 2012
%F A127677 From _Tom Copeland_, Nov 07 2015: (Start)
%F A127677 A logarithmic generator is 2*(1-log(1+x))-log(1-t*x/(1+x)^2) = 2 - log(1+(2-t)*x+x^2) = 2 + (-2 + t)*x + (2 - 4*t + t^2) x^2/2 + (-2 + 9*t - 6*t^2 + t^3) x^3/3 + ..., so a number of relations to the Faber polynomials of A263916 hold with p(0,x) = 2:
%F A127677 1) p(n,x) = F(n,(2-x),1,0,0,..)
%F A127677 2) p(n,x) = (-1)^n 2 + F(n,-x,2x,-3x,...,(-1)^n n*x)
%F A127677 3) p(n,x) = (-1)^n [2 + F(n,x,2x,3x,...,n*x)].
%F A127677 The unsigned array contains the partial sums of A111125 modified by appending a column of zeros, except for an initial two, to A111125. Then the difference of consecutive rows of unsigned A127677, further modified by appending an initial rows of zeros, generates the modified A111125. Cf. A208513 and A034807.
%F A127677 For relations among the characteristic polynomials of Cartan matrices of the Coxeter root groups, Chebyshev polynomials, cyclotomic polynomials, and the polynomials of this entry, see Damianou (p. 12, 20, and 21) and Damianou and Evripidou (p. 7).
%F A127677 See A111125 for a relation to the squares of the odd row polynomials here with the constant removed.
%F A127677 p(n,x)^2 = 2 + p(2*n,x). See also A127672. (End)
%F A127677 a(n,m) = -2*a(n-1,m) + a(n-1,m-1) - a(n-2,m) for n >= 2 with initial conditions a(0,0) = 2, a(1,0) = -2, a(1,1) = 1, a(0,m) = 0 for m != 0, a(1,m) = 0 for m != 0,1. - _William P. Orrick_, Jun 09 2020
%F A127677 p(n,x) = (x-2)*p(n-1,x) - p(n-2,x) for n >= 2. - _William P. Orrick_, Jun 09 2020
%e A127677 The triangle a(n,m) starts:
%e A127677 n\m  0    1    2     3     4     5     6     7    8   9  10 ...
%e A127677 0:   2
%e A127677 1:  -2    1
%e A127677 2:   2   -4    1
%e A127677 3:  -2    9   -6     1
%e A127677 4:   2  -16   20    -8     1
%e A127677 5:  -2   25  -50    35   -10     1
%e A127677 6:   2  -36  105  -112    54   -12     1
%e A127677 7:  -2   49 -196   294  -210    77   -14     1
%e A127677 8:   2  -64  336  -672   660  -352   104   -16    1
%e A127677 9:  -2   81 -540  1386 -1782  1287  -546   135  -18   1
%e A127677 10:  2 -100  825 -2640  4290 -4004  2275  -800  170 -20  1
%e A127677 ... Reformatted and extended by _Wolfdieter Lang_, Nov 21 2012.
%e A127677 n=3: [-2,9,-6,1] stands for -2*1 + 9*(2*x)^2 -6*(2*x)^4 +1*(2*x)^6 = 2*(1+18*x^2-48*x^4+32*x^6) = 2*T(6,x).
%e A127677 (4*n+2)-gon side/radius s(4*n+2) as polynomial in rho(4*n+2) = smallest diagonal/side: n=0: s(2) = 2 (rho(2)=0); n=1: s(6) = -2 + rho(6)^2 = -2 + 3 = 1, (C(6,x) = x^2 - 3); n=2: s(10) = 2 - 4*rho(10)^2 + 1*rho(10)^4 = 2 - 4*rho(10)^2 + (5*rho(10)^2 - 5) = -3 + rho(10)^2, (C(10,x) = x^4 - 5*x^2 + 5). - _Wolfdieter Lang_, Oct 04 2013
%t A127677 T[n_, m_, d_] := If[ n == m, 2, If[n == d && m == d - 1, -2, If[(n == m - 1 || n == m + 1), -1, 0]]] M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}] a = Join[M[1], Table[CoefficientList[CharacteristicPolynomial[M[d], x], x], {d, 1, 10} ]] (* _Roger L. Bagula_, May 23 2007 *)
%t A127677 CoefficientList[2 ChebyshevT[2 Range[0, 10], Sqrt[x]/2], x] // Flatten (* _Eric W. Weisstein_, Apr 06 2017 *)
%t A127677 CoefficientList[Table[(-1)^n LucasL[2 n, Sqrt[-x]], {n, 0, 10}], x] // Flatten (* _Eric W. Weisstein_, Apr 06 2017 *)
%o A127677 (PARI) a(n,m) = {if(n>=2, -2*a(n-1,m)+a(n-1,m-1)-a(n-2,m), if(n==0, if(m!=0,0,2), if(m==0,-2, if(m==1,1,0))))};
%o A127677 for(n=0,10,for(m=0,n,print1(a(n,m),", "))) \\ _Hugo Pfoertner_, Jul 19 2020
%Y A127677 Cf. A284982 (opposite signs and rows begin with 0).
%Y A127677 Row sums (signed): -A061347(n+3) for n>=0.
%Y A127677 Row sums (unsigned): A005248(n) = L(2*n), where L=Lucas.
%Y A127677 Cf. A005248, A053122.
%Y A127677 Cf. A263916, A111125, A208513, A034807, A127672.
%K A127677 sign,tabl,easy
%O A127677 0,1
%A A127677 _Wolfdieter Lang_, Mar 07 2007
%E A127677 Definition corrected by _Eric W. Weisstein_, Apr 06 2017