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URL: https://oeis.org/A334823

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A334823
Triangle, read by rows, of Lambert's denominator polynomials related to convergents of tan(x).
3
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
OFFSET
0,4
COMMENTS
Lambert's numerator polynomials related to convergents of tan(x), g(n, x), are given in A334824.
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, 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).
T(n, 0) = A001147(n).
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
(* Alternative: *)
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]];
(SageMath) [[ 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
Columns k: A001147 (k=0), A001879 (k=2), A001880 (k=4), A038121 (k=6).
Sequence in context: A277410 A368054 A289546 * A279031 A304336 A392435
KEYWORD
tabl,sign
AUTHOR
G. C. Greubel, May 12 2020, following a suggestion from Michel Marcus
STATUS
approved