cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A046988 Numerators of zeta(2*n)/Pi^(2*n).

Original entry on oeis.org

-1, 1, 1, 1, 1, 1, 691, 2, 3617, 43867, 174611, 155366, 236364091, 1315862, 6785560294, 6892673020804, 7709321041217, 151628697551, 26315271553053477373, 308420411983322, 261082718496449122051, 3040195287836141605382, 5060594468963822588186
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Equivalently, numerator of (-1)^(n+1)*2^(2*n-1)*Bernoulli(2*n)/(2*n)!. - Lekraj Beedassy, Jun 26 2003
An old name erroneously included "Numerators of Taylor series expansion of log(x/sin(x))"; that is now submitted as a distinct sequence A283301. - Vladimir Reshetnikov, Mar 04 2017

Examples

			Numerator(zeta(0)/Pi^0) = -1. - _Artur Jasinski_, Mar 11 2010

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.
  • A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 84.

Links

Crossrefs

Cf. A002432 (denominators), A283301, A266214.

Programs

  • Maple
    seq(numer(Zeta(2*n)/Pi^(2*n)),n=1..24); # Martin Renner, Sep 07 2016
  • Mathematica
    Table[Numerator[Zeta[2 n]/Pi^(2 n)], {n, 0, 30}] (* Artur Jasinski, Mar 11 2010 *)

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

Original entry on oeis.org

1, 6, 180, 2835, 37800, 467775, 3831077250, 127702575, 2605132530000, 350813659321125, 15313294652906250, 147926426347074375, 2423034863565078262500, 144228265688397515625, 3952575621190533915703125, 84913182070036240111050234375, 999843529136357459316262500000
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

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]
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.