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A264234
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Numerators of the coefficients in the expansion of 1/W(x) - 1/x where W(x) is the Lambert W function.
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3
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1, -1, 2, -9, 32, -625, 324, -117649, 131072, -4782969, 1562500, -25937424601, 35831808, -23298085122481, 110730297608, -4805419921875, 562949953421312, -48661191875666868481, 91507169819844, -104127350297911241532841, 640000000000000000, -865405750887126927009
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OFFSET
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0,3
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COMMENTS
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If prefixed by an additional 1, numerators of coefficients of expansion of exp(W(x)). - N. J. A. Sloane, Jan 08 2021
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LINKS
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FORMULA
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a(n) = (-1)^n*numerator(g(n)) where g(n) = n^n/n!.
a(n) = (-1)^n*denominator(h(n)) where h(n) = Sum_{k=0..n-1}(n!*n^k)/(k!*n^n).
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EXAMPLE
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Coefficients of expansion of exp(W(x)) are 1, 1, -1/2, 2/3, -9/8, 32/15, -625/144, 324/35, -117649/5760, 131072/2835, -4782969/44800, ... - N. J. A. Sloane, Jan 08 2021
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MAPLE
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seq(numer((-1)^n*n^n/n!), n = 0..21);
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MATHEMATICA
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CoefficientList[Series[1/ProductLog[x] - 1/x, {x, 0, 21}], x] // Numerator
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PROG
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(PARI) vector(22, n, n--; (-1)^n*numerator(n^n/n!)) \\ Altug Alkan, Nov 09 2015
(Magma) [(-1)^n * Numerator(n^n/Factorial(n)): n in [0..50]]; // G. C. Greubel, Nov 14 2017
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CROSSREFS
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KEYWORD
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sign,frac
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AUTHOR
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STATUS
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approved
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