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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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

Views

Author

Wolfdieter Lang, Nov 28 2012

Keywords

Comments

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).

Examples

			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.

Crossrefs

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

Formula

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

A220665 Array of coefficients of powers of x^2 for (S(2*n+1,x)/x)^3, with Chebyshev's S polynomials A049310.

Original entry on oeis.org

1, -8, 12, -6, 1, 27, -108, 171, -136, 57, -12, 1, -64, 480, -1488, 2488, -2472, 1524, -588, 138, -18, 1, 125, -1500, 7575, -21200, 36690, -41700, 32211, -17184, 6330, -1580, 255, -24, 1, -216, 3780, -28098, 117323, -308688, 546864, -680474, 611019, -402264, 195444, -69894, 18153, -3328, 408, -30, 1
Offset: 0

Views

Author

Wolfdieter Lang, Dec 17 2012

Keywords

Comments

The row lengths sequence of this array is 3*n+1 = A016777(n).
For the coefficient array of S(n,x)^3 see A219240. The present array is the odd part of the bisection of that one divided by x^3.
The row polynomials in powers of x^2 are (S(2*n+1,x)/x)^3 = sum(a(n,m)*x^(2*m), m=0..3*n), n >= 0. The o.g.f. for these row polynomials is GS3odd(x,z) = ((z+1)^2 +2*z*(x^2-3))/ (((z+1)^2-z*x^2)*((z+1)^2-z*x^2*(x^2-3)^2)). This is obtained from the odd part of the bisection of the o.g.f. for A219240.

Examples

			The array a(n,m) begins:
n\m  0    1     2     3      4     5     6    7    8  9
0:   1
1:  -8   12    -6     1
2:  27 -108   171  -136     57   -12     1
3: -64  480 -1488  2488  -2472  1524  -588  138  -18  1
...
Row n=4: [125 -1500, 7575, -21200, 36690, -41700, 32211, -17184, 6330, -1580, 255, -24, 1],
Row n=5: [-216, 3780, -28098, 117323, -308688, 546864, -680474, 611019, -402264, 195444, -69894, 18153, -3328, 408, -30, 1],
Row n=6: [343, -8232, 84378, -489608, 1809129, -4562292, 8219967, -10918992, 10927077, -8356272, 4923132, -2240256, 784840, -209580, 41853, -6048, 597, -36, 1],
Row n=1: (S(3,x)/x)^3 = -8 + 12*x^2 - 6*x^4 + 1*x^6, with Chebyshev's S polynomial.

Crossrefs

Cf. A219240, A220666 (even part of the bisection).

Formula

a(n,m) = [x^m](S(2*n+1,x)/x)^3, n>=0, 0 <= m <= 3*n.
a(n,m) = [x^m]([z^n]GS3odd(x,z)) with GS3odd(x,z) the o.g.f. for the row polynomials in powers of x^2, given in a comment above.

A220666 Array of coefficients of powers of x^2 for S(2*n,x)^3 with Chebyshev's S polynomials A049310.

Original entry on oeis.org

1, -1, 3, -3, 1, 1, -9, 30, -45, 30, -9, 1, -1, 18, -123, 399, -651, 588, -308, 93, -15, 1, 1, -30, 345, -1921, 5598, -9540, 10212, -7137, 3303, -1003, 192, -21, 1, -1, 45, -780, 6609, -29847, 80718, -141482, 168927, -141636, 84766, -36366, 11091, -2346, 327, -27, 1
Offset: 0

Views

Author

Wolfdieter Lang, Dec 17 2012

Keywords

Comments

The row lengths sequence of this array is 3*n+1 = A016777(n).
For the coefficient array of S(n,x)^3 see A219240. The present array is the even part of the bisection of that one.
The row polynomials in powers of x^2 are (S(2*n,x))^3 =
sum(a(n,m)*x^(2*m), m=0..3*n), n >= 0. The o.g.f. for these row polynomials is GS3even(x,z) = ((z+1)^3 + (1+z)*z*x^2*(3*x^2 - 7))/(((z+1)^2-z*x^2)*((z+1)^2-z*x^2*(x^2-3)^2)). This is obtained from the even part of the bisection of the o.g.f. for A219240.

Examples

			The array a(n,m) begins:
n\m  0    1     2     3      4     5     6    7    8  9
0:   1
1:  -1    3    -3     1
2:   1   -9    30   -45     30    -9     1
3:   1   18  -123   399   -651   588  -308   93  -15  1
...
Row n=4: [1, -30, 345, -1921, 5598, -9540, 10212, -7137, 3303, -1003, 192, -21, 1],
Row n=5: [-1, 45, -780, 6609, -29847, 80718, -141482, 168927, -141636, 84766, -36366, 11091, -2346, 327, -27, 1],
Row n=6: [1, -63, 1533, -18333, 118029, -460815, 1184872, -2118207, 2729922, -2598297, 1854177, -999687, 407472, -124680, 28164, -4553, 498, -33, 1].
Row n=2: S(4,x)^3 = 1 - 9*x^2 + 30*x^4 - 45*x^6 + 30*x^8  - 9*x^10 + 1*x^12.

Crossrefs

Cf. A219240, A220665 (odd part of the bisection).

Formula

a(n,m) = [x^m] S(2*n,x)^3, n>=0, 0 <= m <= 3*n.
a(n,m) = [x^m]([z^n]GS3even(x,z)) with GS3even(x,z) the o.g.f. for the row polynomials in powers of x^2, given in a comment above.
Showing 1-3 of 3 results.

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