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%I #33 Sep 30 2023 20:58:28
%S 1,2,13,143,1852,27563,473725,9290396,203745235,4912490375,
%T 128777672338,3643086083981,110557605978901,3579776914324250,
%U 123074955978249433,4474133111905169219,171363047274358839412,6893620459732188296591,290475101469031118494993
%N Dowling numbers: e.g.f. exp(x + (exp(b*x)-1)/b) with b=9.
%H Muniru A Asiru, <a href="/A003581/b003581.txt">Table of n, a(n) for n = 0..180</a>
%H Moussa Benoumhani, <a href="http://dx.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(9*x) - 1)/9).
%F G.f.: 1/W(0), where W(k) = 1 - x - x/(1 - 9*(k+1)*x/W(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Nov 07 2014
%F a(n) = exp(-1/9) * Sum_{k>=0} (9*k + 1)^n / (9^k * k!). - _Ilya Gutkovskiy_, Apr 16 2020
%F a(n) ~ 9^(n + 1/9) * n^(n + 1/9) * exp(n/LambertW(9*n) - n - 1/9) / (sqrt(1 + LambertW(9*n)) * LambertW(9*n)^(n + 1/9)). - _Vaclav Kotesovec_, Jun 26 2022
%e G.f. = 1 + 2*x + 13*x^2 + 143*x^3 + 1852*x^4 + 27563*x^5 + ...
%p seq(coeff(series(factorial(n)*exp(z+(1/9)*exp(9*z)-(1/9)),z,n+1), z, n), n = 0 .. 20); # _Muniru A Asiru_, Feb 24 2019
%t With[{m=20, b=9}, 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] * 9^k * BellB[k, 1/9], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Apr 17 2020 *)
%o (PARI) Vec(serlaplace(exp(z + (exp(9*z) - 1)/9))) \\ _Michel Marcus_, Nov 07 2014
%o (Magma) m:=20; c:=9; 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=9; 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), this sequence (b=9), A003582 (b=10), A364069 (b=63), A364070 (b=624).
%K nonn
%O 0,2
%A _N. J. A. Sloane_
%E Name clarified by _Muniru A Asiru_, Feb 24 2019
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