|
|
A334823
|
|
Triangle, read by rows, of Lambert's denominator polynomials related to convergents of tan(x).
|
|
2
|
|
|
1, 1, 0, 3, 0, -1, 15, 0, -6, 0, 105, 0, -45, 0, 1, 945, 0, -420, 0, 15, 0, 10395, 0, -4725, 0, 210, 0, -1, 135135, 0, -62370, 0, 3150, 0, -28, 0, 2027025, 0, -945945, 0, 51975, 0, -630, 0, 1, 34459425, 0, -16216200, 0, 945945, 0, -13860, 0, 45, 0, 654729075, 0, -310134825, 0, 18918900, 0, -315315, 0, 1485, 0, -1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,4
|
|
COMMENTS
|
Lambert's numerator polynomials related to convergents of tan(x), g(n, x), are given in A334824.
|
|
LINKS
|
|
|
FORMULA
|
Equals the coefficients of the polynomials, f(n, x), defined by: (Start)
f(n, x) = Sum_{k=0..floor(n/2)} ((-1)^k*(2*n-2*k)!/((2*k)!*(n-2*k)!))*(x/2)^(n-2*k).
f(n, x) = ((2*n)!/n!)*(x/2)^n*Hypergeometric2F3(-n/2, (1-n)/2; 1/2, -n, -n+1/2; -1/x^2).
f(n, x) = ((-i)^n/2)*(y(n, i*x) + (-1)^n*y(n, -i*x)), where y(n, x) are the Bessel Polynomials.
f(n, x) = (2*n-1)*x*f(n-1, x) - f(n-2, x).
E.g.f. of f(n, x): cos((1 - sqrt(1-2*x*t))/2)/sqrt(1-2*x*t).
f(n, 1) = (-1)^n*f(n, -1) = A053983(n) = (-1)^(n+1)*A053984(-n-1) = (-1)^(n+1) * g(-n-1, 1).
f(n, 2) = (-1)^n*f(n, -2) = A053988(n+1). (End)
As a number triangle:
T(n, k) = i^k*(2*n-k)!*(1+(-1)^k)/(2^(n-k+1)*(k)!*(n-k)!), where i = sqrt(-1).
|
|
EXAMPLE
|
Polynomials:
f(0, x) = 1;
f(1, x) = x;
f(2, x) = 3*x^2 - 1;
f(3, x) = 15*x^3 - 6*x;
f(4, x) = 105*x^4 - 45*x^2 + 1;
f(5, x) = 945*x^5 - 420*x^3 + 15*x;
f(6, x) = 10395*x^6 - 4725*x^4 + 210*x^2 - 1;
f(7, x) = 135135*x^7 - 62370*x^5 + 3150*x^3 - 28*x;
f(8, x) = 2027025*x^8 - 945945*x^6 + 51975*x^4 - 630*x^2 + 1.
Triangle of coefficients begins as:
1;
1, 0;
3, 0, -1;
15, 0, -6, 0;
105, 0, -45, 0, 1;
945, 0, -420, 0, 15, 0;
10395, 0, -4725, 0, 210, 0, -1;
135135, 0, -62370, 0, 3150, 0, -28, 0;
2027025, 0, -945945, 0, 51975, 0, -630, 0, 1.
|
|
MAPLE
|
T:= (n, k) -> I^k*(2*n-k)!*(1+(-1)^k)/(2^(n-k+1)*(k)!*(n-k)!);
seq(seq(T(n, k), k = 0 .. n), n = 0 .. 10);
|
|
MATHEMATICA
|
(* First program *)
y[n_, x_]:= Sqrt[2/(Pi*x)]*E^(1/x)*BesselK[-n -1/2, 1/x];
f[n_, k_]:= Coefficient[((-I)^n/2)*(y[n, I*x] + (-1)^n*y[n, -I*x]), x, k];
Table[f[n, k], {n, 0, 10}, {k, n, 0, -1}]//Flatten
(* Second program *)
Table[ I^k*(2*n-k)!*(1+(-1)^k)/(2^(n-k+1)*(k)!*(n-k)!), {n, 0, 10}, {k, 0, n}]//Flatten
|
|
PROG
|
(Magma)
C<i> := ComplexField();
T:= func< n, k| Round( i^k*Factorial(2*n-k)*(1+(-1)^k)/(2^(n-k+1)*Factorial(k)*Factorial(n-k)) ) >;
[T(n, k): k in [0..n], n in [0..10]];
(Sage) [[ i^k*factorial(2*n-k)*(1+(-1)^k)/(2^(n-k+1)*factorial(k)*factorial(n-k)) for k in (0..n)] for n in (0..10)]
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|