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%I #30 Mar 14 2024 11:06:07
%S 1,2,14,168,2356,37832,701464,14866848,352943376,9219925792,
%T 261954304224,8033968939648,264411579439936,9288709762556032,
%U 346608927301622144,13680000261825018368,569006722158124974336,24864267879086770135552,1138321277772163220033024
%N Dowling numbers: e.g.f. exp(x + (exp(b*x)-1)/b) with b=10.
%C 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
%H Muniru A Asiru, <a href="/A003582/b003582.txt">Table of n, a(n) for n = 0..180</a>
%H Moussa Benoumhani, <a href="https://doi.org/10.1016/0012-365X(95)00095-E">On Whitney numbers of Dowling lattices</a>, Discrete Math. 159 (1996), no. 1-3, 13-33.
%H Paweł Hitczenko, <a href="https://arxiv.org/abs/2403.03422">A class of polynomial recurrences resulting in (n/log n, n/log^2 n)-asymptotic normality</a>, arXiv:2403.03422 [math.CO], 2024. See p. 8.
%F E.g.f.: exp(x + (exp(10*x) - 1)/10).
%F a(n) = exp(-1/10) * Sum_{k>=0} (10*k + 1)^n / (10^k * k!). - _Ilya Gutkovskiy_, Apr 16 2020
%F 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
%p 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
%t 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 *)
%t Table[Sum[Binomial[n, k] * 10^k * BellB[k, 1/10], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Apr 17 2020 *)
%o (PARI) my(x='x+O('x^20)); b=10; Vec(serlaplace(exp(x +(exp(b*x)-1)/b))) \\ _G. C. Greubel_, Feb 24 2019
%o (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
%o (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
%Y Cf. A000110 (b=1), A007405 (b=2), A003575 (b=3), A003576 (b=4), A003577 (b=5), A003578 (b=6), A003579 (b=7), A003580 (b=8), A003581 (b=9), this sequence (b=10).
%K nonn
%O 0,2
%A _N. J. A. Sloane_
%E Name clarified by _Muniru A Asiru_, Feb 24 2019
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