A334824 Triangle, read by rows, of Lambert's numerator polynomials related to convergents of tan(x).

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, 34459425, 0, -4729725, 0, 135135, 0, -990, 0, 1, 654729075, 0, -91891800, 0, 2837835, 0, -25740, 0, 55, 0, 13749310575, 0, -1964187225, 0, 64324260, 0, -675675, 0, 2145, 0, -1

Offset: 0

Author

G. C. Greubel, May 13 2020, following a suggestion from Michel Marcus

Keywords

• tabl
• sign

Comments

Lambert's denominator polynomials related to convergents of tan(x), f(n, x), are given in A334823.

Examples

			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.
		

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.

Crossrefs

Cf. A001497, A001498, A053983, A053984, A053987, A053988, A094675, A334823.

Columns k: A001147 (k=0), A000457 (k=2), A001881 (k=4), A130563 (k=6).

Programs

Magma

C := 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]];

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

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)]

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, 1) = (-1)^n*g(n, -1) = A053984(n) = (-1)^n*A053983(-n-1) = (-1)^n*f(-n-1, 1).

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).