A230284
Denominators to Dirichlet inverse of Euler totient based version of series expansion for x/LambertW(x).
Original entry on oeis.org
1, 1, 2, 3, 8, 15, 144, 35, 5760, 315, 5600, 693, 43545600, 1001, 6706022400, 6435, 14014, 109395, 376610217984000, 46189, 128047474114560000, 323323, 2540395, 2028117, 26976017466662584320000, 96577, 3241475864250624, 35102025, 2126818694000, 5386025
Offset: 1
Similar to but strictly different from
A264235.
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Clear[nn, n, k, s, x]; nn = 22; Denominator[CoefficientList[1 + Integrate[1 + Expand[Sum[Exp[Limit[Zeta[s]*Sum[(If[n == 1, 0, Table[DivisorSum[m, # MoebiusMu[#] &], {m, nn}][[GCD[n, k]]]])/(k)^(s - 1), {k, 1, n}], s -> 1]]*(-x)^n, {n, 1, nn}]], x], x]]
A264234
Numerators of the coefficients in the expansion of 1/W(x) - 1/x where W(x) is the Lambert W function.
Original entry on oeis.org
1, -1, 2, -9, 32, -625, 324, -117649, 131072, -4782969, 1562500, -25937424601, 35831808, -23298085122481, 110730297608, -4805419921875, 562949953421312, -48661191875666868481, 91507169819844, -104127350297911241532841, 640000000000000000, -865405750887126927009
Offset: 0
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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[(-1)^n * Numerator(n^n/Factorial(n)): n in [0..50]]; // G. C. Greubel, Nov 14 2017
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seq(numer((-1)^n*n^n/n!), n = 0..21);
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CoefficientList[Series[1/ProductLog[x] - 1/x, {x, 0, 21}], x] // Numerator
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vector(22, n, n--; (-1)^n*numerator(n^n/n!)) \\ Altug Alkan, Nov 09 2015
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