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URL: https://oeis.org/A334824

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A334824
Triangle, read by rows, of Lambert's numerator polynomials related to convergents of tan(x).
3
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
0,2
COMMENTS
Lambert's denominator polynomials related to convergents of tan(x), f(n, x), are given in A334823.
LINKS
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, 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).
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
(* Alternative: *)
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]];
(SageMath) [[ 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
Columns k: A001147 (k=0), A000457 (k=2), A001881 (k=4), A130563 (k=6).
Sequence in context: A135399 A275831 A065121 * A167339 A277936 A138540
KEYWORD
tabl,sign
AUTHOR
G. C. Greubel, May 13 2020, following a suggestion from Michel Marcus
STATUS
approved