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User: Jonny Griffiths

Jonny Griffiths has authored 5 sequences.

A228356 The triangle associated with the family of polynomials W_n(x).

1, 1, 1, -1, 3, 1, -1, 5, 5, 1, 1, 7, 19, 7, 1, 1, 9, 71, 41, 9, 1, -1, 11, 265, 239, 71, 11, 1, -1, 13, 989, 1393, 559, 109, 13, 1, 1, 15, 3691, 8119, 4401, 1079, 155, 15, 1, 1, 17, 13775, 47321, 34649, 10681, 1847, 209, 17, 1

Offset: 0

Jonny Griffiths, Aug 28 2013

W_n(x) is the family of polynomials in x with integer coefficients given by W_n(x) = sin((2n+1)arccos(x)/2)/(sin(arccos(x)/2)).

These polynomials are intimately linked with the Chebyshev polynomials of the first and second kinds, and represent the polynomials associated with the Dirichlet kernel.

			The triangle is given here as W_0(0)=1, W_1(0)=1, W_0(1)=1, W_2(0)=-1, W_1(1)=3, W_0(2)=1, W_3(0)=-1, W_2(1)=5 ...

Cf. A028297, A101124, A133156, A228161, A228565, A228637, A180870.

W[0, ] = 1; W[1, x] := 2 x + 1; W[n_, x_] := W[n, x] = 2 x W[n - 1, x] - W[n - 2, x]; Table[W[n - x, x] , {n, 0, 9}, {x, 0, n}] // Flatten (* Jean-François Alcover, Jun 11 2017 *)

W_{n+1} = 2xW_n(x) - W_{n-1}, W_0(x)=1, W_1(x)=2x+1.

A228637 The number triangle associated with the polynomials V_n(x).

Original entry on oeis.org

1, -1, 1, -1, 1, 1, 1, 1, 3, 1, 1, 1, 11, 5, 1, -1, 1, 41, 29, 7, 1, -1, 1, 153, 169, 55, 9, 1, 1, 1, 571, 985, 433, 89, 11, 1, 1, 1, 2131, 5741, 3409, 881, 131, 13, 1, -1, 1, 7953, 33461, 26839, 8721, 1561, 181, 15, 1

Offset: 0

Jonny Griffiths, Aug 28 2013

V(n) is the polynomial with integer coefficients in x given by cos((2n+1)(arccos(x)/2))/(arccos(x)/2). The triangle here is given by V_0(0), V_1(0), V_0(1), V_2(0), V_1(1), V_0(2), V_3(0), V_2(1), V_1(2), V_0(3), V_4(0),....

			V_0(x)=1, V_1(x)=2x-1, V_2(x)=4x^2-2x-1,  ...

Cf. A028297, A101124, A133156, A228161, A228565, A180870.

The terms are given by the recurrence relation V_{n+1}(x) = 2xV_n(x)-V_{n-1}(x), V_0(x) = 1, V_1(x)=2x-1.

A228565 Triangle read by rows: coefficients of descending powers of the polynomial V(n,x) = cos((2n+1)(arccos(x)/2))/cos(arccos(x)/2), n >= 0.

Original entry on oeis.org

1, 2, -1, 4, -2, -1, 8, -4, -4, 1, 16, -8, -12, 4, 1, 32, -16, -32, 12, 6, -1, 64, -32, -80, 32, 24, -6, -1, 128, -64, -192, 80, 80, -24, -8, 1, 256, -128, -448, 192, 240, -80, -40, 8, 1, 512, -256, -1024, 448, 672, -240, -160, 40, 10, -1, 1024, -512, -2304, 1024, 1792, -672, -560, 160, 60, -10, -1, 2048, -1024, -5120, 2304, 4608, -1792, -1792, 560, 280, -60, -12, 1, 4096, -2048, -11264, 5120, 11520, -4608, -5376, 1792, 1120, -280, -84, 12, 1

Offset: 0

Jonny Griffiths, Aug 25 2013

V(n,x) is related to the Dirichlet kernel and its associated polynomials. V(n,x) arises in studying recurrences connecting the Chebyshev polynomials of the first and second kinds. It differs from A180870 above only in the signs of terms.

Chebyshev polynomials V(n,x) of the third kind (see, for example, Mason and Handscomb, Chapter 1, Definition 1.3). See A180870 for Chebyshev polynomials of the fourth kind. Cf. A155751. - Peter Bala, Jan 17 2014

			V(0,x) = 1, V(1,x) = 2x-1, V(2,x) = 4x^2-2x-1, V(3,x) = 8x^3 -4x^2 - 4x + 1, V(4,x) = 16x^4 - 8x^3 - 12x^2 + 4x + 1, V(5,x) = 32x^5 - 16x^4 - 32x^3 + 12x^2 + 6x - 1, V(6,x) =64x^6 - 32x^5 - 80x^4 + 32x^3 + 24x^2 - 6x - 1, ...
Triangle begins:
     1;
     2,   -1;
     4,   -2,    -1;
     8,   -4,    -4,    1;
    16,   -8,   -12,    4,    1;
    32,  -16,   -32,   12,    6,   -1;
    64,  -32,   -80,   32,   24,   -6,   -1;
   128,  -64,  -192,   80,   80,  -24,   -8,   1;
   256, -128,  -448,  192,  240,  -80,  -40,   8,   1;
   512, -256, -1024,  448,  672, -240, -160,  40,  10,  -1;
  1024, -512, -2304, 1024, 1792, -672, -560, 160,  60, -10,  -1;
  ...

J. C. Mason and D. C. Handscomb, Chebyshev polynomials, Chapman and Hall/CRC, 2002.

Paul Barry, On the Group of Almost-Riordan Arrays, arXiv preprint arXiv:1606.05077 [math.CO], 2016.

Cf. A180870, A028297. A008312, A155751.

A228565 := proc(n,k)
    local t,Vn,x ;
    t := arccos(x) ;
    Vn := cos((n+1/2)*t)/cos(t/2) ;
    coeftayl(%,x=0,n-k) ;
end proc:
for n from 0 to 10 do
    for k from 0 to n do
        printf("%d,",A228565(n,k)) ;
    end do:
    printf("\n") ;
end do: # R. J. Mathar, Mar 12 2014

V[n_] := Cos[(2*n + 1)*(ArcCos[x]/2)]/Cos[ArcCos[x]/2];
row[n_] := CoefficientList[V[n] + O[x]^(n + 1), x] // Reverse;
Table[row[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Nov 20 2017 *)

V(n+1,x) = 2xV(n,x) - V(n-1,x) with V(0,x) = 1, V(1,x) = 2x-1.
From Peter Bala, Jan 17 2014: (Start)
O.g.f. (1 - t)/(1 - 2*x*t + t^2) = 1 + (2*x - 1)*t +(4*x^2 - 2*x - 1)*t^2 + ....
In terms of the Chebyshev polynomials T(n,x) of the first kind and Chebyshev polynomials U(n,x) of the second kind we have
V(n,x) = U(n,x) - U(n-1,x);
V(n,x) + V(n-1,x) = 2*T(n,x);
V(n,x) = 1/u*T(2*n+1,u) with u = sqrt((1 + x)/2).
Also binomial(2*n,n)*V(n,x) = 2^(2*n)*Jacobi_P(n,-1/2,1/2,x). (End)

A228161 Number triangle associated to Chebyshev polynomials of the second kind.

Original entry on oeis.org

1, 0, 1, -1, 2, 1, 0, 3, 4, 1, 1, 4, 15, 6, 1, 0, 5, 56, 35, 8, 1, -1, 6, 209, 204, 63, 10, 1, 0, 7, 780, 1189, 496, 99, 12, 1, 1, 8, 2911, 6930, 3905, 980, 143, 14, 1, 0, 9, 10864, 40391, 30744, 9701, 1704, 195, 16, 1, -1, 10, 40545, 235416, 242047, 96030, 20305, 2716, 255, 18, 1

Offset: 0

Jonny Griffiths, Aug 14 2013

Compare the definition of U_n(x) with the definition of the Dirichlet kernel.
U_n(x) is defined as sin((n+1)*arccos(x))/sin(arccos(x)).
U_n(x) is a polynomial in x with integer coefficients for all n >=0.
The initial term is U_0(0).
The triangle is given here as U_0(0), U_1(0), U_1(1), U_2(0), U_2(1), U_2(2), U_3(0),....

			Triangle begins:
  1,
  0, 1,
 -1, 2,  1,
  0, 3,  4,  1,
  1, 4, 15,  6, 1,
  0, 5, 56, 35, 8, 1,
  ...

T. D. Noe, Rows n = 0..100 of triangle, flattened
Wikipedia, Chebyshev polynomials

Cf. A101124 (number triangle for Chebyshev polynomials of the first kind).

Cf. A133156 (coefficients of powers of x in U_n(x)).

nn = 10; Flatten[Table[ChebyshevU[i - j, j], {i, 0, nn}, {j, 0, i}]] (* T. D. Noe, Aug 16 2013 *)

The polynomials can be computed with U_{n+1}(x) = 2*x*U_n(x) - U_{n-1}(x), U_{n+1}(x) = ((U_n(x))^2-1)/U_{n-1}(x), where in each case U_0(x) = 1; U_1(x) = 2*x.

More terms from Michel Marcus, Feb 24 2025

A180870 D(n, x) is the Dirichlet kernel sin((n+1/2)x)/sin(x/2). The triangle gives in row n the coefficients of descending powers of x of the polynomial D(n, arccos(x)).

Original entry on oeis.org

1, 2, 1, 4, 2, -1, 8, 4, -4, -1, 16, 8, -12, -4, 1, 32, 16, -32, -12, 6, 1, 64, 32, -80, -32, 24, 6, -1, 128, 64, -192, -80, 80, 24, -8, -1, 256, 128, -448, -192, 240, 80, -40, -8, 1, 512, 256, -1024, -448, 672, 240, -160, -40, 10, 1

Offset: 0

Jonny Griffiths, Sep 21 2010

D(n, arccos(x)) = U(n, x) + U(n-1, x) where U(n, x) are the Chebyshev polynomials of the second kind. These polynomials arise naturally in the investigation of the integer triples (p, q, (p*q + 1)/(p + q)).

Chebyshev polynomials of the fourth kind, usually denoted by W(n, x) (see, for example, Mason and Handscomb, Chapter 1, Definition 1.3). See A228565 for Chebyshev polynomials of the third kind. Cf. A157751. - Peter Bala, Jan 17 2014

			The triangle T(n,m) begins:
n\m    0   1     2     3    4   5    6    7   8  9  10 ...
0:     1
1:     2   1
2:     4   2    -1
3:     8   4    -4    -1
4:    16   8   -12    -4    1
5:    32  16   -32   -12    6   1
6:    64  32   -80   -32   24   6   -1
7:   128  64  -192   -80   80  24   -8   -1
8:   256 128  -448  -192  240  80  -40   -8   1
9:   512 256 -1024  -448  672 240 -160  -40  10  1
10: 1024 512 -2304 -1024 1792 672 -560 -160  60 10  -1
... reformatted - _Wolfdieter Lang_, Jul 26 2014
Recurrence: T(4,2) = (1 + 1)*T(3,2) - T(3,1) = 2*(-4) - 4 = -12. T(4,3) = 0*T(3,3) - (-1)*T(3,2) = T(3,2) = -4. - _Wolfdieter Lang_, Jul 30 2014

J. C. Mason and D. C. Handscomb, Chebyshev polynomials, Chapman and Hall/CRC 2002.

Paul Barry, On the Group of Almost-Riordan Arrays, arXiv preprint arXiv:1606.05077 [math.CO], 2016.
Wikipedia, Dirichlet kernel.

Cf. A008312, A028297, A157751, A228565, A049310, A244419 (row reversed triangle).

ogf := (1 + t)/(1 - 2*x*t + t^2):
ser := simplify(series(ogf, t, 12)): tc := n -> coeff(ser, t, n):
Trow := n -> local k; seq(coeff(tc(n), x, n-k), k = 0..n):
seq(print(Trow(n)), n = 0..9);  # Peter Luschny, Oct 07 2024

row(n) = {if (n==0, return([1])); f = 2*x+1; for (k = 2, n, for (i = 1, (k-1)\2 + 1, f += (-1)^(i+1)*(binomial(k-i, i-1)*(2*x)^(k-2*i+2) - 2*binomial(k-1-i, i-1)*(2*x)^(k-2*i)););); Vec(f);} \\ Michel Marcus, Jul 18 2014

From Peter Bala, Jan 17 2014: (Start)
O.g.f. (1 + t)/(1 - 2*x*t + t^2) = 1 + (2*x + 1)*t + (4*x^2 + 2*x - 1)*t^2 + ....
Recurrence equation: W(0,x) = 1, W(1,x) = 2*x + 1 and W(n,x) = 2*x*W(n-1,x) - W(n-2,x) for n >= 2.
In terms of U(n,x), the Chebyshev polynomials of the second kind, we have W(n,x) = U(2*n,u) with u = sqrt((1 + x)/2). Also binomial(2*n,n)*W(n,x) = 2^(2*n)*Jacobi_P(n,1/2,-1/2,x). (End)
Row sums: 2*n+1. - Michel Marcus, Jul 16 2014
T(n,m) = [x^(n-m)](U(n, x) + U(n-1, x)) = [x^(n-m)] S(2*n, sqrt(2*(1+x))), n >= m >= 0, with U(n, x) = S(n, 2*x). The coefficient triangle of the Chebyshev S-polynomials is given in A049310. See the Peter Bala comments above. - Wolfdieter Lang, Jul 26 2014
From Wolfdieter Lang, Jul 30 2014: (Start)
O.g.f. for the row polynomials R(n,x) = Sum_{m=0..n} T(n,m)*x^m, obtained from the one given by Peter Bala above by row reversion: (1 + x*t)/(1 - 2*t + (x*t)^2).
In analogy to A157751 one can derive a recurrence for the row polynomials R(n, x) = x^n*Dir(n,1/x) with Dir(n,x) = U(n,x) + U(n-1,x) using also negative arguments but only one recursive step: R(n,x) = (1+x)*R(n-1,-x) + R(n-1,x), n >= 1, R(0,x) = 1 (R(-1,x) = -1/x). Proof: derive the o.g.f. and compare it with the known one.
This entails the triangle recurrence T(n,m) = (1 + (-1)^m)* T(n-1,m) - (-1)^m*T(n-1,m-1), for n >= m >= 1 with T(n,m) = 0 if n < m and T(n,0) = 2^n. (End)

Missing term in sequence corrected by Paul Curtz, Dec 31 2011

Edited (name reformulated, Wikipedia link added) by Wolfdieter Lang, Jul 26 2014

cp's OEIS Frontend

User: Jonny Griffiths

A228356 The triangle associated with the family of polynomials W_n(x).

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A228637 The number triangle associated with the polynomials V_n(x).

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A228565 Triangle read by rows: coefficients of descending powers of the polynomial V(n,x) = cos((2n+1)(arccos(x)/2))/cos(arccos(x)/2), n >= 0.

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A228161 Number triangle associated to Chebyshev polynomials of the second kind.

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A180870 D(n, x) is the Dirichlet kernel sin((n+1/2)x)/sin(x/2). The triangle gives in row n the coefficients of descending powers of x of the polynomial D(n, arccos(x)).

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