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Dowling numbers: e.g.f. exp(x + (exp(b*x)-1)/b) with b=5.
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%I #33 Sep 08 2022 08:44:32

%S 1,2,9,63,536,5307,60389,775988,11062391,172638727,2921519374,

%T 53221709973,1037320865141,21517178350762,472862758184789,

%U 10966587174511443,267502464814857936,6842498829509972687,183057455239626138009,5110016898453125496548

%N Dowling numbers: e.g.f. exp(x + (exp(b*x)-1)/b) with b=5.

%H Muniru A Asiru, <a href="/A003577/b003577.txt">Table of n, a(n) for n = 0..440</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(5*x) - 1)/5).

%F a(n) = exp(-1/5) * Sum_{k>=0} (5*k + 1)^n / (5^k * k!). - _Ilya Gutkovskiy_, Apr 16 2020

%F a(n) ~ 5^(n + 1/5) * n^(n + 1/5) * exp(n/LambertW(5*n) - n - 1/5) / (sqrt(1 + LambertW(5*n)) * LambertW(5*n)^(n + 1/5)). - _Vaclav Kotesovec_, Jun 26 2022

%t With[{m=20, b=5}, 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] * 5^k * BellB[k, 1/5], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Apr 17 2020 *)

%o (PARI) my(x='x+O('x^20)); b=5; Vec(serlaplace(exp(x +(exp(b*x)-1)/b))) \\ _G. C. Greubel_, Feb 24 2019

%o (Magma) m:=20; c:=5; 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=5; 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

%o (GAP) b:=5;; a:=[1,2];; for n in [3..20] do a[n]:=2*a[n-1]+Sum([0..n-3],i->Binomial(n-2,i)*b^(n-2-i)*a[i+1]); od; Print(a); # _Muniru A Asiru_, Apr 10 2019

%Y Cf. A000110 (b=1), A007405 (b=2), A003575 (b=3), A003576 (b=4), this sequence (b=5), A003578 (b=6), A003579 (b=7), A003580 (b=8), A003581 (b=9), A003582 (b=10).

%K nonn

%O 0,2

%A _N. J. A. Sloane_

%E Name clarified by _Muniru A Asiru_, Feb 24 2019