A003576 Dowling numbers: e.g.f.: exp(x + (exp(b*x) - 1)/b) with b=4.

1, 2, 8, 48, 352, 3008, 29440, 324096, 3947520, 52541440, 757260288, 11733385216, 194272854016, 3419584921600, 63707979972608, 1251489089060864, 25836869372608512, 558946705406427136, 12638569755079344128, 298003073694026432512, 7312035980392431353856

Offset

0,2

Links

Muniru A Asiru, Table of n, a(n) for n = 0..230

Moussa Benoumhani, On Whitney numbers of Dowling lattices, Discrete Math. 159 (1996), no. 1-3, 13-33.

Formula

E.g.f.: exp(z + (exp(4*z) - 1)/4).

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

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

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

Maple

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

Mathematica

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 *)
Table[Sum[Binomial[n, k] * 4^k * BellB[k, 1/4], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 17 2020 *)

Prog

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

Crossrefs

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).

Sequence in context: A136722 A085615 A054726 * A225042 A326887 A095989

Adjacent sequences: A003573 A003574 A003575 * A003577 A003578 A003579

Keyword

nonn

Author

N. J. A. Sloane

Status

approved