A028297 -id:A028297 - cp's OEIS frontend

A201863 Let CZ(0,x)=1, CZ(1,x)=0 , CZ(2,x)=x^2-1 and CZ(n,x)=2xCZ(n-1,x) - CZ(n-2,x) for n > 2. This sequence is the triangle of polynomial coefficients in order of decreasing exponents.

1, 0, 0, 1, 0, -1, 2, 0, -2, 0, 4, 0, -5, 0, 1, 8, 0, -12, 0, 4, 0, 16, 0, -28, 0, 13, 0, -1, 32, 0, -64, 0, 38, 0, -6, 0, 64, 0, -144, 0, 104, 0, -25, 0, 1, 128, 0, -320, 0, 272, 0, -88, 0, 8, 0, 256, 0, -704, 0, 688, 0, -280, 0, 41, 0, -1

Offset: 0

Paul Curtz, Dec 06 2011

From (A039991 without 0's=) A028297 we wrote in A201509
1,  1,
2,  2,
4,  5, 1,
8, 12, 4.
Hence a(n) first coefficients:
1,
0, 0
1, 0,- 1,   x^2-1,
2, 0, -2, 0,
4, 0, -5, 0, 1
8, 0,-12, 0, 4, 0.
The first 1 is a choice.
Row sums=0.
Absolute value row sums: 1 before A163271.
First vertical:A034008=1 before A131577. Third:-A045623.
Mirror image of triangle in A076626. - Philippe Deléham, Dec 07 2011

Cf. A039991, A201509.

A028298 Triangle of coefficients in expansion of sin(nx) (or sin(nx)/cos(x) if n is even) in ascending powers of sin(x).

Original entry on oeis.org

1, 2, 3, -4, 4, -8, 5, -20, 16, 6, -32, 32, 7, -56, 112, -64, 8, -80, 192, -128, 9, -120, 432, -576, 256, 10, -160, 672, -1024, 512, 11, -220, 1232, -2816, 2816, -1024, 12, -280, 1792, -4608, 5120, -2048, 13, -364, 2912, -9984, 16640, -13312, 4096, 14, -448, 4032, -15360, 28160, -24576, 8192, 15, -560, 6048

Offset: 1

N. J. A. Sloane

Rows have ceiling(n/2) terms.

			Triangle begins:
   1;
   2;
   3,   -4;
   4,   -8;
   5,  -20,  16;
   6,  -32,  32;
   7,  -56, 112,   -64;
   8,  -80, 192,  -128;
   9, -120, 432,  -576, 256;
  10, -160, 672, -1024, 512;
  ...
sin 3x = 3 sin x - 4 sin^3 x;
sin 4x / cos x = 4 sin x - 8 sin^3 x, etc.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, 5th ed., Section 1.335, p. 35.

Cf. A028297.

t[n_] := (Sin[n x]/If[EvenQ[n], Cos[x], 1] // TrigExpand) /. Cos[x]^m_ /; EvenQ[m] -> (1 - Sin[x]^2)^(m/2) // Expand; Flatten[Table[ Partition[ CoefficientList[t[n], Sin[x]] , 2][[All, 2]], {n, 1, 15}]][[1 ;; 59]]  (* Jean-François Alcover, May 06 2011 *)

T(n,k):=(-1)^k*((n-2*k)*(-1)^n-n-2*k)/(2*n+(-1)^n-1+4*k)*2^(2*k+((-1)^n-1)/2)*binomial((2*n+(-1)^n-1)/4+k,(2*n-(-1)^n+1)/4-k); /* Tani Akinari, Jul 15 2024 */

T(n,k) = (-1)^k*((n-2*k)*(-1)^n-n-2*k)/(2*n+(-1)^n-1+4*k)*2^(2*k+((-1)^n-1)/2)*binomial((2*n+(-1)^n-1)/4+k,(2*n-(-1)^n+1)/4-k). - Tani Akinari, Jul 15 2024

More terms from Larry Reeves (larryr(AT)acm.org), Sep 08 2000

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.

A340554 T(n, k) = [x^k] hypergeom([-2^n/2, -2^n/2 - 1/2], [1/2], x). Triangle read by rows, T(n, k) for n >= 0.

Original entry on oeis.org

1, 1, 1, 3, 1, 10, 5, 1, 36, 126, 84, 9, 1, 136, 2380, 12376, 24310, 19448, 6188, 680, 17, 1, 528, 40920, 1107568, 13884156, 92561040, 354817320, 818809200, 1166803110, 1037158320, 573166440, 193536720, 38567100, 4272048, 237336, 5456, 33

Offset: 0

Peter Luschny, Feb 03 2021

			Triangle starts:
                           [0] 1, 1
                           [1] 1, 3
                         [2] 1, 10, 5
                     [3] 1, 36, 126, 84, 9
     [4] 1, 136, 2380, 12376, 24310, 19448, 6188, 680, 17

G. C. Greubel, Rows n = 0..11 of the irregular triangle, flattened

Cf. A001146 (row sums), A000051 (main diagonal), A016131 (central terms), A201461, A028297.

p:= func< n | n eq 0 select 1 else 2^(n-1) >;
T:= func< n,k | Factorial(2^n+1)/(Factorial(2*k)*Factorial(2^n-2*k+1)) >;
[T(n,k): k in [0..p(n)], n in [0..8]]; // G. C. Greubel, Dec 30 2024

CoeffList := p -> op(PolynomialTools:-CoefficientList(p, x)):
Tpoly := proc(n) simplify(hypergeom([-2^n/2, -2^n/2 - 1/2], [1/2], x)):
CoeffList(%) end: seq(Tpoly(n), n = 0..5);

Tpoly[n_] := HypergeometricPFQ[{-2^n/2, -2^n/2 - 1/2}, {1/2}, x];
Table[CoefficientList[Tpoly[n], x], {n, 0, 5}] // Flatten

# from sage.all import * # (use for Python)
def p(n): return 1 if n==0 else pow(2,n-1)
def T(n,k): return rising_factorial(-pow(2,n)-1, 2*k)/factorial(2*k)
print(flatten([[T(n,k) for k in range(p(n)+1)] for n in range(8)])) # G. C. Greubel, Dec 30 2024

T(n, k) = (2^n + 1)!/((2*k)! * (2^n - 2*k + 1)!), for n >= 0, 0 <= k <= p(n), where p(n) = 1 if n = 0 otherwise p(n) = 2^(n-1). Alternative form: T(n, k) = Pochhammer(-2^n - 1, 2*k)/(2*k)!. - G. C. Greubel, Dec 30 2024

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.

cp's OEIS Frontend

A201863 Let CZ(0,x)=1, CZ(1,x)=0 , CZ(2,x)=x^2-1 and CZ(n,x)=2*x*CZ(n-1,x) - CZ(n-2,x) for n > 2. This sequence is the triangle of polynomial coefficients in order of decreasing exponents.

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A028298 Triangle of coefficients in expansion of sin(n*x) (or sin(n*x)/cos(x) if n is even) in ascending powers of sin(x).

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

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A340554 T(n, k) = [x^k] hypergeom([-2^n/2, -2^n/2 - 1/2], [1/2], x). Triangle read by rows, T(n, k) for n >= 0.

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

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Formula

A201863 Let CZ(0,x)=1, CZ(1,x)=0 , CZ(2,x)=x^2-1 and CZ(n,x)=2xCZ(n-1,x) - CZ(n-2,x) for n > 2. This sequence is the triangle of polynomial coefficients in order of decreasing exponents.

A028298 Triangle of coefficients in expansion of sin(nx) (or sin(nx)/cos(x) if n is even) in ascending powers of sin(x).