A101124 -id:A101124 - cp's OEIS frontend

A322836 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) is Chebyshev polynomial of the first kind T_{n}(x), evaluated at x=k.

1, 1, 0, 1, 1, -1, 1, 2, 1, 0, 1, 3, 7, 1, 1, 1, 4, 17, 26, 1, 0, 1, 5, 31, 99, 97, 1, -1, 1, 6, 49, 244, 577, 362, 1, 0, 1, 7, 71, 485, 1921, 3363, 1351, 1, 1, 1, 8, 97, 846, 4801, 15124, 19601, 5042, 1, 0, 1, 9, 127, 1351, 10081, 47525, 119071, 114243, 18817, 1, -1

Offset: 0

Seiichi Manyama, Dec 28 2018

			Square array begins:
   1, 1,    1,     1,      1,      1,       1, ...
   0, 1,    2,     3,      4,      5,       6, ...
  -1, 1,    7,    17,     31,     49,      71, ...
   0, 1,   26,    99,    244,    485,     846, ...
   1, 1,   97,   577,   1921,   4801,   10081, ...
   0, 1,  362,  3363,  15124,  47525,  120126, ...
  -1, 1, 1351, 19601, 119071, 470449, 1431431, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Wikipedia, Chebyshev polynomials.
Index entries for sequences related to Chebyshev polynomials.

Mirror of A101124.
Columns 0-20 give A056594, A000012, A001075, A001541, A001091, A001079, A023038, A011943(n+1), A001081, A023039, A001085, A077422, A077424, A097308, A097310, A068203, A322888, A056771, A322889, A078986, A322890.
Rows 0-10 give A000012, A001477, A056220, A144129, A144130, A243131, A243132, A243133, A243134, A243135, A243136.
Main diagonal gives A115066.
Cf. A323182 (Chebyshev polynomial of the second kind).

Table[ChebyshevT[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Amiram Eldar, Dec 28 2018 *)

T(n,k) = polchebyshev(n,1,k);
matrix(7, 7, n, k, T(n-1,k-1)) \\ Michel Marcus, Dec 28 2018

T(n, k) = round(cos(n*acos(k)));\\ Seiichi Manyama, Mar 05 2021

T(n, k) = if(n==0, 1, n*sum(j=0, n, (2*k-2)^j*binomial(n+j, 2*j)/(n+j))); \\ Seiichi Manyama, Mar 05 2021

A(0,k) = 1, A(1,k) = k and A(n,k) = 2 * k * A(n-1,k) - A(n-2,k) for n > 1.
A(n,k) = cos(n*arccos(k)). - Seiichi Manyama, Mar 05 2021
A(n,k) = n * Sum_{j=0..n} (2*k-2)^j * binomial(n+j,2*j)/(n+j) for n > 0. - Seiichi Manyama, Mar 05 2021

A101125 Row sums of a Chebyshev number triangle.

Original entry on oeis.org

1, 1, 1, 4, 13, 49, 233, 1240, 7201, 45521, 311225, 2285116, 17909309, 149080865, 1312597361, 12180044528, 118740086369, 1212695223137, 12942512039697, 144018843991220, 1667526171728525, 20053044685823697, 250043383489271193

Offset: 0

Paul Barry, Dec 02 2004

Row sums of A101124.

Seiichi Manyama, Table of n, a(n) for n = 0..533
Index entries for sequences related to Chebyshev polynomials.

Table[Sum[ChebyshevT[k, n-k], {k, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Jan 20 2019 *)

{a(n) = sum(k=0, n, polchebyshev(k, 1, n-k))} \\ Seiichi Manyama, Jan 20 2019

a(n)=sum{k=0..n, if(k

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

A101126 Diagonal sums of a Chebyshev number triangle.

Original entry on oeis.org

1, 0, 0, 1, 3, 3, 8, 30, 117, 466, 1960, 8655, 40391, 200085, 1051888, 5844844, 34131049, 208344660, 1324262256, 8743620525, 59891078347, 425216195191, 3126418416440, 23778261543210, 186815245299549, 1513931016854598

Offset: 0

Paul Barry, Dec 02 2004

Diagonal sums of number triangle A101124.

Index entries for sequences related to Chebyshev polynomials.

a(n)=sum{k=0..floor(n/2), if(k

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.

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

Original entry on oeis.org

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.

Showing 1-6 of 6 results.

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A322836 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) is Chebyshev polynomial of the first kind T_{n}(x), evaluated at x=k.

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A101125 Row sums of a Chebyshev number triangle.

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

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A101126 Diagonal sums of a Chebyshev number triangle.

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

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

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