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

0,7

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, ...

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.

Links

Table of n, a(n) for n=0..22.

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]

Example

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 + ...

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}]

Crossrefs

Cf. A046989 (denominators), A046988.

Sequence in context: A001067 A141590 A276594 * A046988 A189683 A029825

Adjacent sequences: A283298 A283299 A283300 * A283302 A283303 A283304

Keyword

nonn,frac,nice

Author

Vladimir Reshetnikov, Mar 04 2017

Status

approved