A046989 Denominators of Taylor series expansion in powers of x^2 of log(x/sin x).

1, 6, 180, 2835, 37800, 467775, 3831077250, 127702575, 2605132530000, 350813659321125, 15313294652906250, 147926426347074375, 2423034863565078262500, 144228265688397515625, 3952575621190533915703125, 84913182070036240111050234375, 999843529136357459316262500000

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• frac
• nice

Comments

For the numerators see A283301.

Examples

			log(x/sin(x)) = 1/6*x^2 + 1/180*x^4 + 1/2835*x^6 + 1/37800*x^8 + 1/467775*x^10 + 691/3831077250*x^12 + ...
		

References

• L. V. Ahlfors, Complex Analysis, McGraw-Hill, 1979, p. 205
• T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 222, series for log(H(x)/x).
• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 88.
• CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 42.
• Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 32, equation 32:6:4 at page 301.

Crossrefs

Cf. A283301 (numerators), A027641/A027642 (Bernoulli).

Programs

Mathematica

Join[{1},Denominator[Take[CoefficientList[Series[Log[x/Sin[x]],{x,0,50}], x],{3,-1,2}]]] (* Harvey P. Dale, Apr 27 2012 *)

Sage

def a(n): return -numerator((n*factorial(2*n))/(2^(2*n-1)*(-1)^n*bernoulli(2*n))) # Ralf Stephan, Apr 01 2015

Formula

log(x/sin(x)) = Sum_{n>0} (2^(2*n-1)*(-1)^(n+1)*B(2*n)/(n*(2*n)!) * x^(2*n)). - Ralf Stephan, Apr 01 2015 [corrected by Roland J. Etienne, Apr 19 2016]