A003164 Numerators of Van der Pol numbers. (Formerly M5262)

1, -1, 1, -1, -1, 1, 1, -1, -37, 111, 177, -177, -2753, 2753, 827, -827, -8386459, 8386459, 28033727, -28033727, -14529522883, 14529522883, 1799010587, -1799010587, -47497385017, 47497385017, 2217167083651, -19954503752859

Offset

0,9

Comments

Named after the Dutch physicist Balthasar van der Pol (1889-1959). - Amiram Eldar, Sep 04 2021

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

B. van der Pol, Smoothing and "unsmoothing", in Mark Kac, Probability and Related Topics in Physical Sciences, New York 1957, pp. 223—235.

Links

T. D. Noe, Table of n, a(n) for n = 0..200

F. T. Howard, Properties of the van der Pol numbers and polynomials, R. Reine Angew. Math., 260 (1973), pp. 35-46.

Formula

E.g.f. for fractions: x^3/( 6*x*(exp(x)+1)-12*(exp(x)-1) ).

Example

1, -1/2, 1/5, -1/20, -1/350, 1/140, 1/1050, -1/300, -37/57750, 111/38500, 177/250250, -177/45500, ... = A003163/A003164.

Mathematica

max = 27; g[x_] = x^3/(6*x*(E^x + 1) - 12*(E^x - 1)); Numerator[ CoefficientList[ Series[ g[x], {x, 0, max}], x]*Range[0, max]!](* Jean-François Alcover, Nov 17 2011, after g.f. *)

Crossrefs

Cf. A003163 (denominators).

Sequence in context: A195316 A176549 A118536 * A304612 A138638 A193105

Adjacent sequences: A003161 A003162 A003163 * A003165 A003166 A003167

Keyword

sign,frac,nice

Author

N. J. A. Sloane

Status

approved