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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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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, Jan., 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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njas
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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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