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

2, -2, 1, 2, -4, 1, -2, 9, -6, 1, 2, -16, 20, -8, 1, -2, 25, -50, 35, -10, 1, 2, -36, 105, -112, 54, -12, 1, -2, 49, -196, 294, -210, 77, -14, 1, 2, -64, 336, -672, 660, -352, 104, -16, 1, -2, 81, -540, 1386, -1782, 1287, -546, 135, -18, 1, 2, -100, 825, -2640, 4290, -4004, 2275, -800, 170, -20, 1

Offset: 0

Author

Wolfdieter Lang, Mar 07 2007

Keywords

• sign
• tabl
• easy

Comments

2*T(2*n,x) = Sum_{m=0..n} a(n,m)*(2*x)^(2*m).

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

Bisection triangle of A127672 (without zero entries, even part). The odd part is ((-1)^(n-m))*A111125(n,m).

If the leading 2 is replaced by a 1 we get the essentially identical sequence A110162. - N. J. A. Sloane, Jun 09 2007

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

From Wolfdieter Lang, Oct 04 2013: (Start)

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).

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)

Examples

			The triangle a(n,m) starts:
n\m  0    1    2     3     4     5     6     7    8   9  10 ...
0:   2
1:  -2    1
2:   2   -4    1
3:  -2    9   -6     1
4:   2  -16   20    -8     1
5:  -2   25  -50    35   -10     1
6:   2  -36  105  -112    54   -12     1
7:  -2   49 -196   294  -210    77   -14     1
8:   2  -64  336  -672   660  -352   104   -16    1
9:  -2   81 -540  1386 -1782  1287  -546   135  -18   1
10:  2 -100  825 -2640  4290 -4004  2275  -800  170 -20  1
... Reformatted and extended by _Wolfdieter Lang_, Nov 21 2012.
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).
(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
		

References

• R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 62
• 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.

Links

• Wolfdieter Lang, First 10 rows and more.
• P. Damianou , On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2014.
• P. Damianou and C. Evripidou, Characteristic and Coxeter polynomials for affine Lie algebras, arXiv preprint arXiv:1409.3956 [math.RT], 2014.
• Yidong Sun, Numerical triangles and several classical sequences, Fib. Quart., Nov. 2005, pp. 359-370. See Table 1.6 (an unsigned version).
• Eric Weisstein's World of Mathematics, Cartan Matrix
• Eric Weisstein's World of Mathematics, Dynkin Diagram

Crossrefs

Cf. A284982 (opposite signs and rows begin with 0).

Row sums (signed): -A061347(n+3) for n>=0.

Row sums (unsigned): A005248(n) = L(2*n), where L=Lucas.

Cf. A005248, A053122.

Cf. A263916, A111125, A208513, A034807, A127672.

Programs

Mathematica

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 *)
CoefficientList[2 ChebyshevT[2 Range[0, 10], Sqrt[x]/2], x] // Flatten (* Eric W. Weisstein, Apr 06 2017 *)
CoefficientList[Table[(-1)^n LucasL[2 n, Sqrt[-x]], {n, 0, 10}], x] // Flatten (* Eric W. Weisstein, Apr 06 2017 *)

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))))};
for(n=0,10,for(m=0,n,print1(a(n,m),", "))) \\ Hugo Pfoertner, Jul 19 2020

Formula

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.

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

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

From Tom Copeland, Nov 07 2015: (Start)

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:

1) p(n,x) = F(n,(2-x),1,0,0,..)

2) p(n,x) = (-1)^n 2 + F(n,-x,2x,-3x,...,(-1)^n n*x)

3) p(n,x) = (-1)^n [2 + F(n,x,2x,3x,...,n*x)].

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.

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).

See A111125 for a relation to the squares of the odd row polynomials here with the constant removed.

p(n,x)^2 = 2 + p(2*n,x). See also A127672. (End)

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

p(n,x) = (x-2)*p(n-1,x) - p(n-2,x) for n >= 2. - William P. Orrick, Jun 09 2020

Extensions

Definition corrected by Eric W. Weisstein, Apr 06 2017