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.
Original entry on oeis.org
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
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.
-
[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
A156645
Triangle T(n, k, m) = b(n,m)/(b(k,m)*b(n-k,m)), where b(n, k) = Product_{j=1..n} (1 - ChebyshevT(j, k+1)^2), b(n, 0) = n!, and m = 2, read by rows.
Original entry on oeis.org
1, 1, 1, 1, 36, 1, 1, 1225, 1225, 1, 1, 41616, 1416100, 41616, 1, 1, 1413721, 1634261476, 1634261476, 1413721, 1, 1, 48024900, 1885939157025, 64069586905104, 1885939157025, 48024900, 1, 1, 1631432881, 2176372249076025, 2511659716192658889, 2511659716192658889, 2176372249076025, 1631432881, 1
Offset: 0
Triangle begins as:
1;
1, 1;
1, 36, 1;
1, 1225, 1225, 1;
1, 41616, 1416100, 41616, 1;
1, 1413721, 1634261476, 1634261476, 1413721, 1;
1, 48024900, 1885939157025, 64069586905104, 1885939157025, 48024900, 1;
-
b:= func< n, k | n eq 0 select 1 else k eq 0 select Factorial(n) else (&*[1 - Evaluate(ChebyshevT(j), k+1)^2 : j in [1..n]]) >;
T:= func< n,k,m | b(n,m)/(b(k,m)*b(n-k,m)) >;
[T(n,k,2): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 03 2021
-
(* First program *)
b[n_, k_]:= b[n,k]= If[k==0, n!, Product[1 -ChebyshevT[j, k+1]^2, {j,n}]];
T[n_, k_, m_]= b[n,m]/(b[k,m]*b[n-k,m]);
Table[T[n, k, 2], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jul 03 2021 *)
(* Second program *)
T[n_, k_, m_]:= T[n, k, m]= Product[(1 - ChebyshevT[2*j+2*k, m+1])/(1 - ChebyshevT[2*j, m+1]), {j, n-k}];
Table[T[n,k,2], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 03 2021 *)
-
def b(n, k): return factorial(n) if (k==0) else product( 1 - chebyshev_T(j, k+1)^2 for j in (1..n) )
def T(n, k, m): return b(n,m)/(b(k,m)*b(n-k,m))
flatten([[T(n, k, 2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 03 2021
A156646
Triangle T(n, k, m) = b(n,m)/(b(k,m)*b(n-k,m)), where b(n, k) = Product_{j=1..n} (1 - ChebyshevT(j, k+1)^2), b(n, 0) = n!, and m = 10, read by rows.
Original entry on oeis.org
1, 1, 1, 1, 484, 1, 1, 233289, 233289, 1, 1, 112444816, 54198633636, 112444816, 1, 1, 54198168025, 12591535188240100, 12591535188240100, 54198168025, 1, 1, 26123404543236, 2925290638056514680225, 1409984043580226203632400, 2925290638056514680225, 26123404543236, 1
Offset: 0
Triangle begins as:
1;
1, 1;
1, 484, 1;
1, 233289, 233289, 1;
1, 112444816, 54198633636, 112444816, 1;
1, 54198168025, 12591535188240100, 12591535188240100, 54198168025, 1;
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b:= func< n, k | n eq 0 select 1 else k eq 0 select Factorial(n) else (&*[1 - Evaluate(ChebyshevT(j), k+1)^2 : j in [1..n]]) >;
T:= func< n,k,m | b(n,m)/(b(k,m)*b(n-k,m)) >;
[T(n,k,10): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 03 2021
-
(* First program *)
b[n_, k_]:= b[n,k]= If[k==0, n!, Product[1 -ChebyshevT[j, k+1]^2, {j,n}]];
T[n_, k_, m_]= b[n,m]/(b[k,m]*b[n-k,m]);
Table[T[n,k,10], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jul 03 2021 *)
(* Second program *)
T[n_, k_, m_]:= T[n, k, m]= Product[(1 - ChebyshevT[2*j+2*k, m+1])/(1 - ChebyshevT[2*j, m+1]), {j, n-k}];
Table[T[n,k,12], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 03 2021 *)
-
def b(n, k): return factorial(n) if (k==0) else product( 1 - chebyshev_T(j, k+1)^2 for j in (1..n) )
def T(n, k, m): return b(n,m)/(b(k,m)*b(n-k,m))
flatten([[T(n, k, 10) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 03 2021
A173585
Triangle T(n, k, q) = c(n, q)/(c(k, q)*c(n-k, q)), where c(n, q) = Product_{j=1..n} t(2*j, q), t(n, q) = (1/4)*( (2 + sqrt(q))^n + (2 - sqrt(q))^n - 2 ), and q = 3, read by rows.
Original entry on oeis.org
1, 1, 1, 1, 16, 1, 1, 225, 225, 1, 1, 3136, 44100, 3136, 1, 1, 43681, 8561476, 8561476, 43681, 1, 1, 608400, 1660970025, 23150231104, 1660970025, 608400, 1, 1, 8473921, 322220846025, 62555239000969, 62555239000969, 322220846025, 8473921, 1
Offset: 0
Triangle begins as:
1;
1, 1;
1, 16, 1;
1, 225, 225, 1;
1, 3136, 44100, 3136, 1;
1, 43681, 8561476, 8561476, 43681, 1;
1, 608400, 1660970025, 23150231104, 1660970025, 608400, 1;
-
b:= func< n, k | n eq 0 select 1 else k eq 0 select Factorial(n) else (&*[1 - Evaluate(ChebyshevT(j), k+1)^2 : j in [1..n]]) >;
T:= func< n,k,m | b(n,m)/(b(k,m)*b(n-k,m)) >;
[T(n,k,1): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 06 2021
-
(* First program *)
f[n_, q_]:= (1/4)*((2+Sqrt[q])^n + (2-Sqrt[q])^n -2);
c[n_, q_]:= Product[f[k, q], {k, 2, n, 2}]//Simplify;
T[n_, k_, q_]:= c[n, q]/(c[k, q]*c[n - k, q]);
Table[T[n, k, 3], {n, 0, 10, 2}, {k, 0, n, 2}]//Flatten (* modified by G. C. Greubel, Jul 06 2021 *)
(* Second program *)
t[n_, q_]:= (1/4)*(Round[(2+Sqrt[q])^n + (2-Sqrt[q])^n] -2);
c[n_, q_]:= Product[t[2*j, q], {j,n}];
T[n_, k_, q_]:= c[n, q]/(c[k, q]*c[n-k, q]);
Table[T[n, k, 3], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 06 2021 *)
-
@CachedFunction
def f(n,q): return (1/4)*( round((2 + sqrt(q))^n + (2 - sqrt(q))^n) - 2 )
def c(n,q): return product( f(2*j, q) for j in (1..n))
def T(n,k,q): return c(n, q)/(c(k, q)*c(n-k, q))
flatten([[T(n,k,3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 06 2021
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