A123588 -id:A123588 - cp's OEIS frontend

A123583 Triangle read by rows: T(n, k) is the coefficient of x^k in the polynomial 1 - T_n(x)^2, where T_n(x) is the n-th Chebyshev polynomial of the first kind.

0, 1, 0, -1, 0, 0, 4, 0, -4, 1, 0, -9, 0, 24, 0, -16, 0, 0, 16, 0, -80, 0, 128, 0, -64, 1, 0, -25, 0, 200, 0, -560, 0, 640, 0, -256, 0, 0, 36, 0, -420, 0, 1792, 0, -3456, 0, 3072, 0, -1024, 1, 0, -49, 0, 784, 0, -4704, 0, 13440, 0, -19712, 0, 14336, 0, -4096

Offset: 0

Gary W. Adamson and Roger L. Bagula, Nov 12 2006

All row sum are zero. Row sums of absolute values are in A114619. - Klaus Brockhaus, May 29 2009

			First few rows of the triangle are:
  0;
  1, 0,  -1;
  0, 0,   4, 0,   -4;
  1, 0,  -9, 0,   24, 0,  -16;
  0, 0,  16, 0,  -80, 0,  128, 0,   -64;
  1, 0, -25, 0,  200, 0, -560, 0,   640, 0, -256;
  0, 0,  36, 0, -420, 0, 1792, 0, -3456, 0, 3072, 0, -1024;
First few polynomials (p(n, x) = 1 - T_{n}(x)^2) are:
  p(0, x) = 0,
  p(1, x) = 1 -    x^2,
  p(2, x) = 0    4*x^2 -   4*x^4,
  p(3, x) = 1 -  9*x^2 +  24*x^4 -   16*x^6,
  p(4, x) = 0   16*x^2 -  80*x^4 +  128*x^6 -   64*x^8,
  p(5, x) = 1 - 25*x^2 + 200*x^4 -  560*x^6 +  640*x^8 -  256*x^10,
  p(6, x) = 0   36*x^2 - 420*x^4 + 1792*x^6 - 3456*x^8 + 3072*x^10 - 1024*x^12.

G. C. Greubel, Rows n = 0..50, flattened
Gareth Jones and David Singerman, Belyi Functions, Hypermaps and Galois Groups, Bull. London Math. Soc., 28 (1996), 561-590.
Yuri Matiyasevich, Generalized Chebyshev polynomials.
G. B. Shabat and I. A. Voevodskii, Drawing curves over number fields, The Grothendieck Festschift, vol. 3, Birkhäuser, 1990, 199-227.
G. B. Shabat and A. Zvonkin, Plane trees and algebraic numbers, Contemporary Math., 1994, vol. 178, 233-275.

Cf. A123588, A156647.

[0] cat &cat[ Coefficients(1-ChebyshevT(n)^2): n in [1..8] ];

(* First program *)
Table[CoefficientList[1 - ChebyshevT[n, x]^2, x], {n, 0, 10}]//Flatten
(* Second program *)
T[n_, k_]:= T[n, k]= SeriesCoefficient[(1 -ChebyshevT[2*n,x])/2, {x,0,k}];
Table[T[n, k], {n,0,12}, {k,0,2*n}]//Flatten (* G. C. Greubel, Jul 02 2021 *)

v=[]; for(n=0, 8, v=concat(v, vector(2*n+1, j, polcoeff(1-poltchebi(n)^2, j-1)))); v

def T(n): return ( (1 - chebyshev_T(2*n, x))/2 ).full_simplify().coefficients(sparse=False)
flatten([T(n) for n in (0..12)]) # G. C. Greubel, Jul 02 2021

T(n, k) = coefficients of ( 1 - ChebyshevT(n, x)^2 ).

T(n, k) = coefficients of ( (1 - ChebyshevT(2*n, x))/2 ). - G. C. Greubel, Jul 02 2021

Edited by N. J. A. Sloane, Mar 09 2008

A082649 Triangle of coefficients in expansion of sinh^2(n*x) in powers of sinh(x).

Original entry on oeis.org

1, 4, 4, 16, 24, 9, 64, 128, 80, 16, 256, 640, 560, 200, 25, 1024, 3072, 3456, 1792, 420, 36, 4096, 14336, 19712, 13440, 4704, 784, 49, 16384, 65536, 106496, 90112, 42240, 10752, 1344, 64, 65536, 294912, 552960, 559104, 329472, 114048, 22176, 2160, 81, 262144, 1310720, 2785280, 3276800, 2329600

Offset: 1

Gary W. Adamson, May 16 2003, suggested by Herb Conn

Using arcsin(x) = Pi/2 - arccos(x), valid for -1 < = x <= 1, we find sin^2(k*arcsin(x)) = sin^2(k*arccos(x)) for k odd, while sin^2(k*arcsin(x)) = 1 - sin^2(k*arccos(x)) for k even. Thus the expansion of sin^2(n*x) in powers of cos(x) will produce a similar table of coefficients. See the example section below. - Peter Bala, Feb 02 2017

			sinh^2 x = sinh^2 x
sinh^2 2x = 4 sinh^4 x + 4 sinh^2 x
sinh^2 3x = 16 sinh^6 x + 24 sinh^4 x + 9 sinh^2 x
sinh^2 4x = 64 sinh^8 x + 128 sinh^6 x + 80 sinh^4 x + 16 sinh^2 x
sinh^2 5x = 256 sinh^10 x + 640 sinh^8 x + 560 sinh^6 x + 200 sinh^4 x + 25 sinh^2 x
From _Peter Bala_, Feb 02 2016: (Start)
sin^2(x) = 1 - cos^2(x);
sin^2(2*x) = -4*cos^4(x) + 4*cos^2(x);
sin^2(3*x) = 1 - (16*cos^6(x) - 24*cos^4(x) + 9*cos^2(x));
sin^2(4*x) = -64*cos^8(x) + 128*cos^6(x) - 80*cos^4(x) + 16*cos^2(x);
sin^2(5*x) = 1 - (256*cos^10(x) - 640*cos^8(x) + 560*cos^6(x) - 200*cos^4(x) + 25*cos^2(x)). (End)

Robert Israel, Table of n, a(n) for n = 1..10011 (rows 0 to 140, flattened)

A001108 gives row sums.

Closely related to A123583 and A123588.

g:= (1+x*y)/((1-x*y)*(1-(4+2*y)*x+x^2*y^2)):
S:= series(g,x,15):
seq(seq(coeff(coeff(S,x,n),y,k),k=0..n),n=0..14); # Robert Israel, Dec 20 2017

Table[Reverse[CoefficientList[1/x TrigExpand[Sinh[n ArcSinh[Sqrt[x]]]^2], x]], {n, 7}] // Flatten (* Eric W. Weisstein, Apr 05 2017 *)
Abs[Table[CoefficientList[x^n Piecewise[{{1 - ChebyshevT[n, 1/Sqrt[x]]^2, Mod[n, 2] == 0}, {ChebyshevT[n, 1/Sqrt[x]]^2, Mod[n, 2] == 1}}], x], {n, 10}]] // Flatten (* Eric W. Weisstein, Apr 05 2017 *)

Coefficients are: 4^(n-1), (2n)4^(n-2), (2n)(2n-3)4^(n-3)/2!, (2n)(2n-4)(2n-5)4^(n-4)/3!, (2n)(2n-5)(2n-6)(2n-7)4^(n-5)/4!, (2n)(2n-6)(2n-7)(2n-8)(2n-9)4^(n-6)/5!...

G.f. as triangle: (1+x*y)/((1-x*y)*(1-(4+2*y)*x+x^2*y^2)). - Robert Israel, Dec 20 2017

More terms from Robert Israel, Dec 20 2017

A138331 a(n) = C(n+5, 5)(n+3)(-1)^(n+1)*16/3.

Original entry on oeis.org

-16, 128, -560, 1792, -4704, 10752, -22176, 42240, -75504, 128128, -208208, 326144, -495040, 731136, -1054272, 1488384, -2062032, 2808960, -3768688, 4987136, -6517280, 8419840, -10764000, 13628160, -17100720, 21280896, -26279568, 32220160, -39239552

Offset: 0

Klaus Brockhaus, Mar 15 2008

Fourth column of the triangle defined in A123588, seventh column of the triangle defined in A123583.

Robert Israel, Table of n, a(n) for n = 0..10000
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (-7,-21,-35,-35,-21,-7,-1).

Cf. A007318 (Pascal's triangle), A123588, A123583, A040977.

[ Binomial(n+5, 5)*(n+3)*(-1)^(n+1)*16/3: n in [0..28] ];

k:=3; [ Coefficients(1-ChebyshevT(n+k)^2)[2*k+1]: n in [0..28] ];

seq(binomial(n+5, 5)*(n+3)*(-1)^(n+1)*16/3, n=0..40); # Robert Israel, Oct 26 2017

LinearRecurrence[{-7,-21,-35,-35,-21,-7,-1},{-16,128,-560,1792,-4704,10752,-22176},30] (* Harvey P. Dale, May 27 2017 *)

for(n=0,28,print1(polcoeff(taylor(16*(x-1)/(x+1)^7,x),n),","));

a(n) = coefficient of x^6 in the polynomial 1 - T_(n+3)(x)^2, where T_n(x) is the n-th Chebyshev polynomial of the first kind.
G.f.: 16*(x-1)/(x+1)^7.
a(n) = (-1)^(n+1)*16*A040977(n).
a(n) = a(-n-5). - Bruno Berselli, Sep 13 2011

A334009 Triangle read by rows: T(n, k) = binomial(n + k - 1, 2k - 1) 4^(k - 1) * n/k, 1 <= k <= n.

Original entry on oeis.org

1, 4, 4, 9, 24, 16, 16, 80, 128, 64, 25, 200, 560, 640, 256, 36, 420, 1792, 3456, 3072, 1024, 49, 784, 4704, 13440, 19712, 14336, 4096, 64, 1344, 10752, 42240, 90112, 106496, 65536, 16384, 81, 2160, 22176, 114048, 329472, 559104, 552960, 294912, 65536, 100

Offset: 1

Michael Somos, Apr 12 2020

Let P(n, x) := Sum_{k=1..n} T(n, k)*x^k. Then P(n, P(m, x)) = P(n*m, x) for all n and m in Z.

The r=4 case of the Logistic Map is 4*x*(1 - x) = -P(1, -x). The r=2 case leads to A193862.

			First four rows:
.1
.4...4
.9..24..16
16..80.128..64

Cf. A001108, A039598, A123588, A193862.

T[ n_, k_] := If[k == 0, 0, Binomial[n + k - 1, 2 k - 1] 4^(k - 1) n / k];

{T(n, k) = if(k == 0, 0, binomial(n + k - 1, 2*k - 1) * 4^(k - 1) * n/k)};

P(n, x) = sinh(n * arcsinh(sqrt(x)))^2 = (hypergeom([-n, n], [1/2], -x) - 1)/2 are the row polynomials.
G.f.: Sum_{n, m} T(n, k) * x^k * y^n = x * y * (1 + y) / ((1 - y) * (1 - (2 + 4*x)*y + y^2)).
Row sums are A001108.
T(n, k) = (-1)^n * (-4)^(k-1) * A039598(-k - 1, n - 1) for all n in Z if k<0.
T(n, k) = -(-1)^(n+k) * A123588(n,k) if 1 <= k <= n.

A138332 C(n+7, 7)(n+4)(-1)^(n+1)*16.

Original entry on oeis.org

-64, 640, -3456, 13440, -42240, 114048, -274560, 604032, -1235520, 2379520, -4356352, 7637760, -12899328, 21085440, -33488640, 51845376, -78450240, 116290944, -169206400, 242070400, -341003520, 473616000, -649284480, 879465600, -1178049600

Offset: 0

Klaus Brockhaus, Mar 15 2008

sign

Fifth column of the triangle defined in A123588, ninth column of the triangle defined in A123583.

Index entries for sequences related to Chebyshev polynomials.

Cf. A007318 (Pascal's triangle), A123588, A123583, A053347.

[ Binomial(n+7, 7)*(n+4)*(-1)^(n+1)*16: n in [0..24] ];

k:=4; [ Coefficients(1-ChebyshevT(n+k)^2)[2*k+1]: n in [0..24] ];

for(n=0,24,print1(polcoeff(taylor(64*(x-1)/(x+1)^9,x),n),","));

a(n) = coefficient of x^8 in the polynomial 1 - T_(n+4)(x)^2, where T_n(x) is the n-th Chebyshev polynomial of the first kind.
G.f.: 64*(x-1)/(x+1)^9.
a(n) = (-1)^(n+1)*64*A053347(n).

A138333 C(n+9, 9)(n+5)(-1)^(n+1)*256/5.

Original entry on oeis.org

-256, 3072, -19712, 90112, -329472, 1025024, -2818816, 7028736, -16180736, 34850816, -70946304, 137592832, -255836672, 458422272, -794962432, 1338884096, -2196606720, 3519493120, -5519205120, 8487198720, -12819206400, 19045678080, -27869287680

Offset: 0

Klaus Brockhaus, Mar 15 2008

sign

Sixth column of the triangle defined in A123588, eleventh column of the triangle defined in A123583.

Index entries for sequences related to Chebyshev polynomials.

Cf. A007318 (Pascal's triangle), A123588, A123583, A054334.

[ Binomial(n+9, 9)*(n+5)*(-1)^(n+1)*256/5: n in [0..22] ];

k:=5; [ Coefficients(1-ChebyshevT(n+k)^2)[2*k+1]: n in [0..22] ];

for(n=0,22,print1(polcoeff(taylor(256*(x-1)/(x+1)^11,x),n),","));

a(n) = coefficient of x^10 in the polynomial 1 - T_(n+5)(x)^2, where T_n(x) is the n-th Chebyshev polynomial of the first kind.
G.f.: 256*(x-1)/(x+1)^11.
a(n) = (-1)^(n+1)*256*A054334(n).

A138334 C(n+11, 11)(n+6)(-1)^(n+1)*512/3.

Original entry on oeis.org

-1024, 14336, -106496, 559104, -2329600, 8200192, -25346048, 70606848, -180590592, 429977600, -963149824, 2046693376, -4153583616, 8094162944, -15214592000, 27690557440, -48952949760, 84293314560, -141710499840, 233076480000

Offset: 0

Klaus Brockhaus, Mar 15 2008

sign

Seventh column of the triangle defined in A123588, thirteenth column of the triangle defined in A123583.

Index entries for sequences related to Chebyshev polynomials.

Cf. A007318 (Pascal's triangle), A123588, A123583.

[ Binomial(n+11, 11)*(n+6)*(-1)^(n+1)*512/3: n in [0..19] ];

k:=6; [ Coefficients(1-ChebyshevT(n+k)^2)[2*k+1]: n in [0..19] ];

Table[Binomial[n+11,11](n+6)(-1)^(n+1) 512/3,{n,0,20}] (* Harvey P. Dale, Jun 03 2021 *)

for(n=0,19,print1(polcoeff(taylor(1024*(x-1)/(x+1)^13,x),n),","));

a(n) = coefficient of x^12 in the polynomial 1 - T_(n+6)(x)^2, where T_n(x) is the n-th Chebyshev polynomial of the first kind.

G.f.: 1024*(x-1)/(x+1)^13.

A373504 Triangular array: row n gives the coefficients T(n,k) of powers x^(2k) in the series expansion of ((b^n + b^(-n))/2)^2, where b = x + sqrt(x^2 + 1).

Original entry on oeis.org

1, 1, 1, 1, 4, 4, 1, 9, 24, 16, 1, 16, 80, 128, 64, 1, 25, 200, 560, 640, 256, 1, 36, 420, 1792, 3456, 3072, 1024, 1, 49, 784, 4704, 13440, 19712, 14336, 4096, 1, 64, 1344, 10752, 42240, 90112, 106496, 65536, 16384, 1, 81, 2160, 22176, 114048, 329472, 559104, 552960, 294912, 65536

Offset: 0

Clark Kimberling, Aug 03 2024

Related to Chebyshev polynomials of the first kind; see A123588.

			First 8 rows:
  1
  1    1
  1    4     4
  1    9    24     16
  1   16    80    128     64
  1   25   200    560    640    256
  1   36   420   1792   3456   3072   1024
  1   49   784   4704  13440  19612  14336  4096
The 4th polynomial is 1 + 9 x^2 + 24 x^4 + 16 x^6.

Cf. A000012 (col 0), A000290 (col 1), A002415 ((1/4)*col(2)), A112742 (col 2), A000302 (T(n,n)), A123588, A008310.
Row sums give A055997(n+1).
Triangle without column 0 gives A334009.

p:= proc(n) option remember; (b-> series(
      ((b^n+b^(-n))/2)^2, x, 2*n+1))(x+sqrt(x^2+1))
    end:
T:= (n, k)-> coeff(p(n), x, 2*k):
seq(seq(T(n,k), k=0..n), n=0..10);  # Alois P. Heinz, Aug 03 2024

t[n_] := ((x + Sqrt[x^2 + 1])^n + (x + Sqrt[x^2 + 1])^(-n))/2
u = Expand[Table[FullSimplify[Expand[t[n]]], {n, 0, 10}]^2]
v = Column[CoefficientList[u, x^2]] (* array *)
Flatten[v] (* sequence *)
T[n_, k_] := If[k==0, 1, 4^(k - 1)*(2*Binomial[n + k, 2*k] - Binomial[n + k -1, 2*k -1])]; Flatten[Table[T[n,k],{n,0,9},{k,0,n}]] (* Detlef Meya, Aug 11 2024 *)

T(n, k) = if (k=0) then 1, otherwise 4^(k - 1)*(2*binomial(n + k, 2*k) - binomial(n + k - 1, 2*k - 1)). - Detlef Meya, Aug 11 2024

cp's OEIS Frontend

A123583 Triangle read by rows: T(n, k) is the coefficient of x^k in the polynomial 1 - T_n(x)^2, where T_n(x) is the n-th Chebyshev polynomial of the first kind.

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A082649 Triangle of coefficients in expansion of sinh^2(n*x) in powers of sinh(x).

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A138331 a(n) = C(n+5, 5)*(n+3)*(-1)^(n+1)*16/3.

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A334009 Triangle read by rows: T(n, k) = binomial(n + k - 1, 2*k - 1) * 4^(k - 1) * n/k, 1 <= k <= n.

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A138332 C(n+7, 7)*(n+4)*(-1)^(n+1)*16.

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A138333 C(n+9, 9)*(n+5)*(-1)^(n+1)*256/5.

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A138334 C(n+11, 11)*(n+6)*(-1)^(n+1)*512/3.

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A138331 a(n) = C(n+5, 5)(n+3)(-1)^(n+1)*16/3.

A334009 Triangle read by rows: T(n, k) = binomial(n + k - 1, 2k - 1) 4^(k - 1) * n/k, 1 <= k <= n.

A138332 C(n+7, 7)(n+4)(-1)^(n+1)*16.

A138333 C(n+9, 9)(n+5)(-1)^(n+1)*256/5.

A138334 C(n+11, 11)(n+6)(-1)^(n+1)*512/3.