A283301 Numerators of coefficients at even powers in Taylor series expansion of log(x/sin(x)).

0, 1, 1, 1, 1, 1, 691, 2, 3617, 43867, 174611, 155366, 236364091, 1315862, 3392780147, 6892673020804, 7709321041217, 151628697551, 26315271553053477373, 308420411983322, 261082718496449122051, 3040195287836141605382, 2530297234481911294093

Offset: 0

Author

Vladimir Reshetnikov, Mar 04 2017

Keywords

• nonn
• frac
• nice

Comments

This sequence shares many terms with A046988 (and appears to have been erroneously confused with it), but actually differs from it at indexes 0, 14, 22, 26, 28, 30, 38, 42, 44, 46, 50, 52, 54, 56, 58, 60, ...

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. A046989 (denominators), A046988.

Programs

Mathematica

a[0] = 0; a[n_] := Numerator[((-1)^(n + 1) 2^(2 n - 1) BernoulliB[2 n])/(n (2 n)!)]; Table[a[n], {n, 0, 20}] (* or *)
Numerator@Table[SeriesCoefficient[Log[x/Sin[x]], {x, 0, 2n}], {n, 0, 20}]

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]