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%I #26 Sep 08 2022 08:44:32
%S 1,2,10,80,772,8648,111592,1631360,26518672,472528160,9139219360,
%T 190461416192,4250569655872,101040920561792,2546488866632320,
%U 67772341398044672,1898177372174512384,55780954727160472064,1715291443214323558912,55062161002484359565312
%N Dowling numbers: e.g.f. exp(x + (exp(b*x) - 1)/b), with b=6.
%H Muniru A Asiru, <a href="/A003578/b003578.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.
%F E.g.f.: exp(x + (exp(6*x) - 1)/6).
%F a(n) = exp(-1/6) * Sum_{k>=0} (6*k + 1)^n / (6^k * k!). - _Ilya Gutkovskiy_, Apr 16 2020
%F a(n) ~ 6^(n + 1/6) * n^(n + 1/6) * exp(n/LambertW(6*n) - n - 1/6) / (sqrt(1 + LambertW(6*n)) * LambertW(6*n)^(n + 1/6)). - _Vaclav Kotesovec_, Jun 26 2022
%p seq(coeff(series(factorial(n)*exp(z+(1/6)*exp(6*z)-(1/6)),z,n+1), z, n), n = 0 .. 20); # _Muniru A Asiru_, Feb 23 2019
%t With[{nn=20},CoefficientList[Series[Exp[x+Exp[6x]/6-1/6],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Jun 12 2017 *)
%t Table[Sum[Binomial[n, k] * 6^k * BellB[k, 1/6], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Apr 17 2020 *)
%o (PARI) my(x='x+O('x^20)); b=6; Vec(serlaplace(exp(x +(exp(b*x)-1)/b))) \\ _G. C. Greubel_, Feb 24 2019
%o (Magma) m:=20; c:=6; 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=6; 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), this sequence (b=6), A003579 (b=7), A003580 (b=8), A003581 (b=9), A003582 (b=10).
%K nonn
%O 0,2
%A _N. J. A. Sloane_
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