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A003575 Dowling numbers: e.g.f.: exp(x + (exp(b*x) - 1)/b) with b=3.

%I A003575 #55 Mar 14 2024 11:10:50
%S A003575 1,2,7,35,214,1523,12349,112052,1120849,12219767,143942992,1819256321,
%T A003575 24526654381,350974470746,5308470041299,84554039118383,
%U A003575 1413794176669942,24745966692370607,452277149756692105,8612255652371171012,170517319084490074405
%N A003575 Dowling numbers: e.g.f.: exp(x + (exp(b*x) - 1)/b) with b=3.
%C A003575 Named after the American mathematician Thomas Allan Dowling (b. 1941). - _Amiram Eldar_, Jun 06 2021
%H A003575 Muniru A Asiru, <a href="/A003575/b003575.txt">Table of n, a(n) for n = 0..210</a>
%H A003575 Moussa Benoumhani, <a href="http://dx.doi.org/10.1016/0012-365X(95)00095-E">On Whitney numbers of Dowling lattices</a>, Discrete Math., Vol. 159, No. 1-3 (1996), pp. 13-33.
%H A003575 Thomas A. Dowling, <a href="https://doi.org/10.1016/S0095-8956(73)80007-3">A class of geometric lattices based on finite groups</a>, Journal of Combinatorial Theory, Series B, Vol. 14, No. 1 (1973), pp. 61-86.
%H A003575 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.
%H A003575 Mahid M. Mangontarum and Jacob Katriel, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Mangontarum/mango2.html">On q-Boson Operators and q-Analogues of the r-Whitney and r-Dowling Numbers</a>, J. Int. Seq., Vol. 18 (2015), Article 15.9.8.
%F A003575 E.g.f.: exp(x + (exp(3*x) - 1)/3).
%F A003575 G.f.: 1/(1-x*Q(0)), where Q(k) = 1 + x/(1 - x + 3*x*(k+1)/(x - 1/Q(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 19 2013
%F A003575 a(n) = exp(-1/3) * Sum_{k>=0} (3*k + 1)^n / (3^k * k!). - _Ilya Gutkovskiy_, Apr 16 2020
%F A003575 a(n) ~ 3^(n + 1/3) * n^(n + 1/3) * exp(n/LambertW(3*n) - n - 1/3) / (sqrt(1 + LambertW(3*n)) * LambertW(3*n)^(n + 1/3)). - _Vaclav Kotesovec_, Jun 26 2022
%p A003575 seq(coeff(series(n!*exp(z+(1/3)*exp(3*z)-(1/3)),z,n+1), z, n), n=0..30); # _Muniru A Asiru_, Feb 19 2019
%t A003575 With[{nn=20},CoefficientList[Series[Exp[x+Exp[3x]/3-1/3],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Jan 04 2019 *)
%t A003575 Table[Sum[Binomial[n, k] * 3^k * BellB[k, 1/3], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Apr 17 2020 *)
%o A003575 (PARI) my(x = 'x + O('x^30)); Vec(serlaplace(exp(x + exp(3*x)/3 - 1/3))) \\ _Michel Marcus_, Feb 09 2018
%o A003575 (Magma) m:=30; c:=3; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x+(Exp(c*x)-1)/c) )); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, Feb 20 2019
%o A003575 (Sage)
%o A003575 b=3;
%o A003575 def A003575_list(prec):
%o A003575     P.<x> = PowerSeriesRing(QQ, prec)
%o A003575     return P( exp(x +(exp(b*x)-1)/b) ).egf_to_ogf().list()
%o A003575 A003575_list(30) # _G. C. Greubel_, Feb 20 2019
%Y A003575 Cf. A000110 (b=1), A007405 (b=2), this sequence (b=3), A003576 (b=4), A003577 (b=5), A003578 (b=6), A003579 (b=7), A003580 (b=8), A003581 (b=9), A003582 (b=10).
%K A003575 nonn
%O A003575 0,2
%A A003575 _N. J. A. Sloane_
%E A003575 Name clarified by _G. C. Greubel_, Feb 20 2019