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Search: id:A001662
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| A001662 |
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Numerators in expansion of W(exp(x)) about x=1, where W is the Lambert function.
(Formerly M4896 N2098)
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+0
3
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| 1, 1, 1, -1, -1, 13, -47, -73, 2447, -16811, -15551, 1726511, -18994849, 10979677, 2983409137, -48421103257, 135002366063, 778870772857, -232033147779359, 1305952009204319, 58740282660173759, -1862057132555380307, 16905219421196907793
(list; graph; listen)
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OFFSET
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0,6
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COMMENT
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Coefficients of Airey's converging factor.
(-1)^n times the polynomials with coefficients in triangle A008517, evaluated at -1. - Ralf Stephan, Dec 13 2004
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
J. R. Airey, The "converging factor" in asymptotic series and the calculation of Bessel, Laguerre and other function, Phil. Mag., 24 (1937), 521-552 [ gives 22 terms ].
J. C. P. Miller, A method for the determination of converging factors ..., Proc. Camb. Phil. Soc., 48 (1952), 243-254.
F. D. Murnaghan, Airey's converging factor, Proc. Nat. Acad. Sci. USA, 69 (1972), 440-441.
F. D. Murnaghan and J. W. Wrench, Jr., The Converging Factor for the Exponential Integral, Report 1535, David Taylor Model Basin, U.S. Dept. of Navy, 1963 [ gives first 67 terms ].
P. Wynn, Converging factors for the Weber parabolic cylinder functions of complex argument, Proc. Konin. Ned. Akad. Weten., Series A, 66 (1963), 721-754 (two parts).
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LINKS
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M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210.
J. M. Borwein and R. M. Corless, Emerging tools for experimental mathematics, Amer. Math. Monthly, 106 (No. 10, 1999), 889-909.
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FORMULA
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Let b(n) = a(n)/2^n; then the e.g.f. B(x)=Sum b(n)x^n/n! satisfies exp B(x) = 1 + 2x - B(x).
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EXAMPLE
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W(exp(x)) = 1 +(x-1)/2+(x-1)^2/16-(x-1)^3/192-...
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MAPLE
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series(LambertW(x), x=1, 45);
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PROGRAM
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(PARI) a(n)=if(n<1, !n, n!*4^n/2*polcoeff(serreverse(x+log(1+x+x*O(x^n))), n))
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CROSSREFS
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Cf. A051711.
Sequence in context: A116476 A035340 A127305 this_sequence A031390 A113943 A004467
Adjacent sequences: A001659 A001660 A001661 this_sequence A001663 A001664 A001665
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KEYWORD
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sign,easy,nice,frac
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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More terms from James A. Sellers (sellersj(AT)math.psu.edu), Dec 07 1999
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