A003576 - OEIS

%I #32 Sep 08 2022 08:44:32

%S 1,2,8,48,352,3008,29440,324096,3947520,52541440,757260288,

%T 11733385216,194272854016,3419584921600,63707979972608,

%U 1251489089060864,25836869372608512,558946705406427136,12638569755079344128,298003073694026432512,7312035980392431353856

%N Dowling numbers: e.g.f.: exp(x + (exp(b*x) - 1)/b) with b=4.

%H Muniru A Asiru, <a href="/A003576/b003576.txt">Table of n, a(n) for n = 0..230</a>

%H Moussa Benoumhani, <a href="https://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(z + (exp(4*z) - 1)/4).

%F G.f.: 1/Q(0), where Q(k) = 1 - 2*x*(2*k+1) - 2*x^2*(2*k+2)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Sep 26 2013

%F a(n) = exp(-1/4) * Sum_{k>=0} (4*k + 1)^n / (4^k * k!). - _Ilya Gutkovskiy_, Apr 16 2020

%F a(n) ~ 4^(n + 1/4) * n^(n + 1/4) * exp(n/LambertW(4*n) - n - 1/4) / (sqrt(1 + LambertW(4*n)) * LambertW(4*n)^(n + 1/4)). - _Vaclav Kotesovec_, Jun 26 2022

%p seq(coeff(series(factorial(n)*exp(z+(1/4)*exp(4*z)-(1/4)),z,n+1), z, n), n = 0 .. 20); # _Muniru A Asiru_, Feb 22 2019

%t With[{m=20, b=4}, CoefficientList[Series[Exp[x+(Exp[b*x]-1)/b], {x,0,m}], x]*Range[0, m]!] (* _G. C. Greubel_, Feb 22 2019 *)

%t Table[Sum[Binomial[n, k] * 4^k * BellB[k, 1/4], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Apr 17 2020 *)

%o (PARI) my(x='x+O('x^20)); b=4; Vec(serlaplace(exp(x+(exp(b*x)-1)/b))) \\ _G. C. Greubel_, Feb 22 2019

%o (Magma) m:=20; c:=4; 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 22 2019

%o (Sage) m = 20; b=4; 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 22 2019

%Y Cf. A000110 (b=1), A007405 (b=2), A003575 (b=3), this sequence (b=4), A003577 (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_