OFFSET
0,2
COMMENTS
Lambert's denominator polynomials related to convergents of tan(x), f(n, x), are given in A334823.
LINKS
G. C. Greubel, Rows n = 0..100 of the triangle, flattened
J.-H. Lambert, Mémoire sur quelques propriétés remarquables des quantités transcendantes et logarithmiques (Memoir on some properties that can be traced from circular transcendent and logarithmic quantities), Histoire de l’Académie royale des sciences et belles-lettres (1761), Berlin. See also.
FORMULA
Equals the coefficients of the polynomials, g(n, x), defined by: (Start)
g(n, x) = Sum_{k=0..floor(n/2)} ((-1)^k*(2*n-2*k+1)!/((2*k+1)!*(n-2*k)!))*(x/2)^(n-2*k).
g(n, x) = ((2*n+1)!/n!)*(x/2)^n*Hypergeometric2F3(-n/2, (1-n)/2; 3/2, -n, -n-1/2; -1/x^2).
g(n, x) = ((-i)^n/2)*(y(n+1, i*x) + (-1)^n*y(n+1, -i*x)), where y(n, x) are the Bessel Polynomials.
g(n, x) = (2*n-1)*x*g(n-1, x) - g(n-2, x).
E.g.f. of g(n, x): sin((1 - sqrt(1-2*x*t))/2)/sqrt(1-2*x*t).
g(n, 2) = (-1)^n*g(n, -2) = A053987(n+1). (End)
As a number triangle:
T(n, k) = i^k*(2*n-k+1)!*(1+(-1)^k)/(2^(n-k+1)*(k+1)!*(n-k)!), where i = sqrt(-1).
T(n, 0) = A001147(n+1).
EXAMPLE
Polynomials:
g(0, x) = 1;
g(1, x) = 3*x;
g(2, x) = 15*x^2 - 1;
g(3, x) = 105*x^3 - 10*x;
g(4, x) = 945*x^4 - 105*x^2 + 1;
g(5, x) = 10395*x^5 - 1260*x^3 + 21*x;
g(6, x) = 135135*x^6 - 17325*x^4 + 378*x^2 - 1;
g(7, x) = 2027025*x^7 - 270270*x^5 + 6930*x^3 - 36*x.
Triangle of coefficients begins as:
1;
3, 0;
15, 0, -1;
105, 0, -10, 0;
945, 0, -105, 0, 1;
10395, 0, -1260, 0, 21, 0;
135135, 0, -17325, 0, 378, 0, -1;
2027025, 0, -270270, 0, 6930, 0, -36, 0.
MAPLE
T:= (n, k) -> I^k*(2*n-k+1)!*(1+(-1)^k)/(2^(n-k+1)*(k+1)!*(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];
g[n_, k_]:= Coefficient[((-I)^n/2)*(y[n+1, I*x] + (-1)^n*y[n+1, -I*x]), x, k];
Table[g[n, k], {n, 0, 10}, {k, n, 0, -1}]//Flatten
(* Second program *)
Table[I^k*(2*n-k+1)!*(1+(-1)^k)/(2^(n-k+1)*(k+1)!*(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+(-1)^k)/(2^(n-k+1)*Factorial(k+1)*Factorial(n-k)) ) >;
[T(n, k): k in [0..n], n in [0..10]];
(Sage) [[ i^k*factorial(2*n-k+1)*(1+(-1)^k)/(2^(n-k+1)*factorial(k+1)*factorial(n-k)) for k in (0..n)] for n in (0..10)]
CROSSREFS
KEYWORD
tabl,sign
AUTHOR
G. C. Greubel, May 13 2020, following a suggestion from Michel Marcus
STATUS
approved