A158454 -id:A158454 - cp's OEIS frontend

A217476 Coefficient triangle for the square of the monic integer Chebyshev T-polynomials A127672.

4, 0, 1, 4, -4, 1, 0, 9, -6, 1, 4, -16, 20, -8, 1, 0, 25, -50, 35, -10, 1, 4, -36, 105, -112, 54, -12, 1, 0, 49, -196, 294, -210, 77, -14, 1, 4, -64, 336, -672, 660, -352, 104, -16, 1, 0, 81, -540, 1386, -1782, 1287, -546, 135, -18, 1, 4, -100, 825, -2640, 4290, -4004, 2275, -800, 170, -20, 1

Offset: 0

Wolfdieter Lang, Oct 17 2012

The monic integer T-polynomials, called R(n,x) (in Abramowitz-Stegun C(n,x)), with their coefficient triangle given in A127672, when squared, become polynomials in y=x^2:
  R(n,x)^2 = sum(T(n,k)*y^k,m=0..n).
R(n,x)^2 = 2 + R(2*n,x). From the bisection of the R-(or T-)polynomials, the even part. Directly from the R(m*n,x)=R(m,R(n,x)) property for m=2.
The o.g.f. is G(z,y) := sum((R(n,sqrt(y))^2)*z^n ,n=0..infinity) = (4 + (4 - 3*y)*z + y*z^2)/((1 +(2-y)*z + z^2)*(1-z)). From the bisection.
The o.g.f.s of the columns k>=1 are x^k*(1-x)/(1+x)^(2*k+1),
and for k=0 the o.g.f. is 4/(1-x^2).
Hetmaniok et al. (2015) refer to these as "modified Chebyshev" polynomials. - N. J. A. Sloane, Sep 13 2016

			The triangle begins:
n\k 0    1    2      3     4      5     6     7    8   9  10
0:  4
1:  0    1
2:  4   -4    1
3:  0    9   -6      1
4:  4  -16   20     -8     1
5:  0   25  -50     35   -10      1
6:  4  -36  105   -112    54    -12     1
7:  0   49 -196    294  -210     77   -14     1
8:  4  -64  336   -672   660   -352   104   -16    1
9:  0   81 -540   1386 -1782   1287  -546   135  -18   1
10: 4 -100  825  -2640  4290  -4004  2275  -800  170 -20   1
...
n=2:  R(2,x) = -2 + y, R(2,x)^2 = 4 -4*y + y^2, with y=x^2.
n=3:  R(3,x) = 3*x - x^3, R(3,x)^2 = 9*y - 6*y^2 +y^3, with y=x^2.
T(4,1) = 8*(-1)^3*binomial(5,3)/5 = -16.
T(4,0) = 2 + 8*(-1)^4*binomial(4,4)/4 = 4.
T(n,1) = (-1)^(n-1)*2*n*(n+1)!/((n-1)!*2!*(n+1)) = -((-1)^n)*n^2 = A162395(n), n >= 1.
T(n,2) = (-1)^n*A002415(n), n >= 0.
T(n,3) = -(-1)^n*A040977(n-3), n >= 3.
T(n,4) = (-1)^n*A053347(n-4), n >= 4.
T(n,5) = -(-1)^n*A054334(n-5), n >= 5.

E Hetmaniok, P Lorenc, S Damian, et al., Periodic orbits of boundary logistic map and new kind of modified Chebyshev polynomials in R. Witula, D. Slota, W. Holubowski (eds.), Monograph on the Occasion of 100th Birthday Anniversary of Zygmunt Zahorski. Wydawnictwo Politechniki Slaskiej, Gliwice 2015, pp. 325-343.

Cf. A127672, A158454 (square of S-polynomials), A128495 (sum of square of S-polynomials).

T(n,k) = [x^(2*k)]R(n,x)^2, with R(n,x) the monic integer version of the Chebyshev T(n,x) polynomial.

T(n,k) = 0 if n=1. ([k=0] means 1 if k=0 else 0).

A181878 Coefficient array for square of Chebyshev S-polynomials.

Original entry on oeis.org

1, 1, 1, -2, 1, 4, -4, 1, 1, -6, 11, -6, 1, 9, -24, 22, -8, 1, 1, -12, 46, -62, 37, -10, 1, 16, -80, 148, -128, 56, -12, 1, 1, -20, 130, -314, 367, -230, 79, -14, 1, 25, -200, 610, -920, 771, -376, 106, -16, 1, 1, -30, 295, -1106, 2083, -2232, 1444, -574, 137, -18, 1, 36, -420, 1897, -4352, 5776, -4744, 2486, -832, 172, -20, 1, 1, -42, 581, -3108, 8518, -13672, 13820, -9142, 4013, -1158, 211, -22, 1

Offset: 0

Wolfdieter Lang, Dec 22 2010

For the coefficients of Chebyshev polynomials S(n,x) see A049310.
The row length sequence for this array is A109613 = {1,1,3,3,5,5,...}.
The row polynomials (in x^2) for even row numbers are
  S(2*k,x)^2 = Sum_{m=0..2*k} a(2*k,m)*x^(2*m), k >= 0.
For odd row numbers the row polynomials (in x^2) are
  (S(2*k+1,x)^2)/x^2 = Sum_{m=0..2*k} a(2*k+1,m)*x^(2*m), k >= 0.
The o.g.f. for the polynomials S(n,x)^2 is
  S(x,z):=((1+z)/(1-z))/(1 + (2-x^2)z +z^2). See the link for a proof. Therefore the coefficients constitute the Riordan array (1/(1-x^2),x/(1+x)^2) found as A158454.
The o.g.f. for S(2*k,sqrt(x))^2 is
  (1-2*(1-x)*z+z^2)/((1-z)*(1 - (2-4*x+x^2)*z + z^2)).
The o.g.f. for (S(2*k+1,sqrt(x))^2)/x is
  ((1+z)/(1-z))/(1 - (2-4*x+x^2)*z + z^2).
The row sums A011655(n+1) are the same as those for the triangle A158454.
The alternating row sums for even numbered rows (-1)^n*A007598(n+1) coincide with those of triangle A158454. For odd row numbers n=2k+1 these sums are A049684(k+1), k >= 0 (squares of even-indexed Fibonacci numbers).

			The irregular triangle a(n,m) begins:
  n\m  0    1    2      3     4      5     6    7   8   9  10 ...
  0:   1
  1:   1
  2:   1   -2    1
  3:   4   -4    1
  4:   1   -6   11     -6     1
  5:   9  -24   22     -8     1
  6:   1  -12   46    -62    37    -10     1
  7:  16  -80  148   -128    56    -12     1
  8:   1  -20  130   -314   367   -230    79  -14   1
  9:  25 -200  610   -920   771   -376   106  -16   1
  10:  1  -30  295  -1106  2083  -2232  1444 -574 137 -18   1
  ... Reformatted and extended by _Wolfdieter Lang_, Nov 24 2012

Wolfdieter Lang, First ten rows with more details and proofs.

Cf. A158454, A129818.

Join[{{1}, {1}}, CoefficientList[Table[ChebyshevU[n, Sqrt[x]/2]^2, {n, 2, 10}], x]] // Flatten (* Eric W. Weisstein, Apr 04 2018 *)
Join[{{1}, {1}}, CoefficientList[ChebyshevU[Range[2, 10], Sqrt[x]/2]^2, x]]  // Flatten (* Eric W. Weisstein, Apr 04 2018 *)

a(2*k,m) = (-1)^m*Sum_{j=0..k} binomial(2*k+m-1-2*j, 2*m-1), k >= 0.
a(2*k+1,m) = (-1)^m*Sum_{j=0..k} binomial(2*k+1+m-2*j, 2*m+1), k >= 0.
This derives from the formula for the entries of the Riordan array A158454.
For the o.g.f.s see the comment.

Corrected by Wolfdieter Lang, Jan 21 2011

A219240 Coefficient array for the cube of Chebyshev's S polynomials.

Original entry on oeis.org

1, 0, 0, 0, 1, -1, 0, 3, 0, -3, 0, 1, 0, 0, 0, -8, 0, 12, 0, -6, 0, 1, 1, 0, -9, 0, 30, 0, -45, 0, 30, 0, -9, 0, 1, 0, 0, 0, 27, 0, -108, 0, 171, 0, -136, 0, 57, 0, -12, 0, 1, -1, 0, 18, 0, -123, 0, 399, 0, -651, 0, 588, 0, -308, 0, 93, 0, -15, 0, 1, 0, 0, 0, -64, 0, 480, 0, -1488, 0, 2488, 0, -2472, 0, 1524, 0, -588, 0, 138, 0, -18, 0, 1

Offset: 0

Wolfdieter Lang, Dec 12 2012

The row lengths sequence is 3*n+1 = A016777(n).
For the coefficient triangle for Chebyshev's S polynomials see A049310.
The o.g.f. for S(n,x)^3, n >= 0, is GS(3;x,z) = (1+z^2+2*z*x)/ ((1+z^2-z*x)*(1+z^2-z*x*(x^2-3))). This is obtained from the de Moivre-Binet formula for S(n,x) and the binomial theorem.
In general the monic integer Chebyshev polynomial tau(n,x):= R(2*n+1,x)/x enters, where R(n,x) = 2*T(n,x/2) with Chebyshev's T polynomial (for R see A127672), and the coefficient triangle for tau is given in A111125 (here for the third power of S only tau(0,x) = 1 and tau(1,x) = x^2 - 3 enter).

			The array a(n,m) begins:
n\m   0  1  2  3  4    5   6    7  8    9 10  11 12  13 14 15
n=0:  1
n=1:  0  0  0  1
n=2: -1  0  3  0 -3    0   1
n=3:  0  0  0 -8  0   12   0  -6   0    1
n=4:  1  0 -9  0 30    0 -45   0  30    0 -9   0  1
n=5:  0  0  0 27  0 -108   0  171  0 -136  0  57  0 -12  0  1
...
Row n=6: [-1, 0, 18, 0, -123, 0, 399, 0, -651, 0, 588, 0, -308, 0, 93, 0, -15, 0, 1],
Row n=7: [0, 0, 0, -64, 0, 480, 0, -1488, 0, 2488, 0, -2472, 0, 1524, 0, -588, 0, 138, 0, -18, 0, 1],
Row n=8: [1, 0, -30, 0, 345, 0, -1921, 0, 5598, 0, -9540, 0, 10212, 0, -7137, 0, 3303, 0, -1003, 0, 192, 0, -21, 0, 1].
n=2: S(2,x)^3 = (x^2 - 1)^3 = -1 + 3*x^2 - 3*x^4 + x^6.
n=3: S(3,x)^3 = (x^3 - 2*x)^3 = -8*x^3 + 12*x^5 - 6*x^7 + x^9.

Cf. A049310, A127672, A158454 (square of S polynomials), A219234 (fourth power of S polynomials).

a(n,m) = [x^m] S(n, x)^3, n >= 0, 0 <= m <= 3*n, with Chebyshev's S polynomials (see A049310).
a(n,m) = [x^m]([z^n] GS(3;x,z)), with the o.g.f. GS(3;x,z) given above in a comment.
The row polynomials p(n, x) := Sum_{m=0..3*n} a(n,m)*x^m = S(n, x)^3 are (S(3*n+2, x) - 3*S(n, x))/(x^2 - 4). For the factorization of S polynomials see comments on A049310. - Wolfdieter Lang, Apr 09 2018

A219234 Coefficient array for the fourth power of Chebyshev's S-polynomials as a function of x^2.

Original entry on oeis.org

1, 0, 0, 1, 1, -4, 6, -4, 1, 0, 0, 16, -32, 24, -8, 1, 1, -12, 58, -144, 195, -144, 58, -12, 1, 0, 0, 81, -432, 972, -1200, 886, -400, 108, -16, 1, 1, -24, 236, -1228, 3678, -6612, 7490, -5532, 2701, -864, 174, -20, 1, 0, 0, 256, -2560, 11136, -27776, 44176, -47232, 34912, -18048, 6504, -1600, 256, -24, 1

Offset: 0

Wolfdieter Lang, Nov 28 2012

The row lengths sequence for this array is 2*n+1, given in A005408.
The coefficient triangle for the monic Chebyshev S-polynomials S(n,x) = U(n,x/2) are given in A049310.
The coefficients for S(n,x)^2 are given in A158454 and in A181878 (odd numbered rows shifted by one unit to the left).

			The irregular triangle a(n, m) starts:
n\m  0   1   2     3    4     5    6     7    8    9  10  11 12
0:   1
1:   0   0   1
2:   1  -4   6    -4    1
3:   0   0  16   -32   24    -8    1
4:   1 -12  58  -144  195  -144   58   -12    1
5:   0   0  81  -432  972 -1200  886  -400  108  -16   1
6:   1 -24 236 -1228 3678 -6612 7490 -5532 2701 -864 174 -20  1
...
Row n=7: [0, 0, 256, -2560, 11136, -27776, 44176, -47232, 34912, -18048, 6504, -1600, 256, -24, 1].
Row n=8: [1, -40, 660, -5828, 30194, -96780, 203374, -293464, 300231, -222112, 119938, -47244, 13415, -2672, 354, -28, 1].
Row n=1 polynomial p(1,x) = 1*x^2 = S(1,sqrt(x))^4 = (sqrt(x))^4.
Row n=2 polynomial p(2,x) = 1 - 4*x + 6*x^2 - 4*x^3 + 1*x^4 =
  S(2,sqrt(x))^4 = (-1+x)^4.

Cf. A049310, A127672, A158454, A181878, A219240.

a(n, m) = [x^(2*m)] S(n, x)^4, n >= 0, with the monic Chebyshev S-polynomials given in terms of the U-polynomials in a comment above.
The o.g.f. GS4(x, z) := sum((S(n, x)^4)*z^n,n=0..infinity) = ((1+z)/(1-z))*(1 - (2-3*x^2)*z + z^2)/((1-z*(-2+x^2)+z^2)*(1-z*(2-4*x^2+x^4)+z^2)). For the o.g.f. of the row polynomials p(n,x) :=sum(a(n,m)*x^m,m=0..n) take GS4(sqrt(x), z).
The row polynomial p(n, x^2) = Sum_{m=0..2*n} a(n, m)*x^(2*m) = (S(n, x))^4 = (R(4*(n+1), x) - 4*R(2*(n+1), x) + 6)/(x^2 - 4)^2, where R are the monic Chebyshev T polynomials with coefficients given in A127672. For factorizations of the S polynomials see comments on A049310. - Wolfdieter Lang, Apr 09 2018

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A217476 Coefficient triangle for the square of the monic integer Chebyshev T-polynomials A127672.

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A181878 Coefficient array for square of Chebyshev S-polynomials.

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A219240 Coefficient array for the cube of Chebyshev's S polynomials.

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A219234 Coefficient array for the fourth power of Chebyshev's S-polynomials as a function of x^2.

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