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A003582
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Dowling numbers: e.g.f. exp(x + (exp(b*x)-1)/b) with b=10.
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13
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1, 2, 14, 168, 2356, 37832, 701464, 14866848, 352943376, 9219925792, 261954304224, 8033968939648, 264411579439936, 9288709762556032, 346608927301622144, 13680000261825018368, 569006722158124974336, 24864267879086770135552, 1138321277772163220033024
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OFFSET
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0,2
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COMMENTS
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In general, for b > 0, if e.g.f. = exp(x + (exp(b*x) - 1)/b), then a(n) ~ b^(n + 1/b) * n^(n + 1/b) * exp(n/LambertW(b*n) - n - 1/b) / (sqrt(1 + LambertW(b*n)) * LambertW(b*n)^(n + 1/b)). - Vaclav Kotesovec, Jun 26 2022
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LINKS
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FORMULA
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E.g.f.: exp(x + (exp(10*x) - 1)/10).
a(n) = exp(-1/10) * Sum_{k>=0} (10*k + 1)^n / (10^k * k!). - Ilya Gutkovskiy, Apr 16 2020
a(n) ~ 10^(n + 1/10) * n^(n + 1/10) * exp(n/LambertW(10*n) - n - 1/10) / (sqrt(1 + LambertW(10*n)) * LambertW(10*n)^(n + 1/10)). - Vaclav Kotesovec, Jun 26 2022
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MAPLE
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seq(coeff(series(factorial(n)*exp(z+(1/10)*exp(10*z)-(1/10)), z, n+1), z, n), n = 0 .. 20); # Muniru A Asiru, Feb 24 2019
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MATHEMATICA
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With[{m=20, b=10}, CoefficientList[Series[Exp[x +(Exp[b*x]-1)/b], {x, 0, m}], x]*Range[0, m]!] (* G. C. Greubel, Feb 24 2019 *)
Table[Sum[Binomial[n, k] * 10^k * BellB[k, 1/10], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 17 2020 *)
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PROG
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(PARI) my(x='x+O('x^20)); b=10; Vec(serlaplace(exp(x +(exp(b*x)-1)/b))) \\ G. C. Greubel, Feb 24 2019
(Magma) m:=20; c:=10; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x +(Exp(c*x)-1)/c) )); [Factorial(n-1)*b[n]: n in [1..m-1]]; // G. C. Greubel, Feb 24 2019
(Sage) m = 20; b=10; T = taylor(exp(x + (exp(b*x) -1)/b), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, Feb 24 2019
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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