A003581 Dowling numbers: e.g.f. exp(x + (exp(b*x)-1)/b) with b=9.

1, 2, 13, 143, 1852, 27563, 473725, 9290396, 203745235, 4912490375, 128777672338, 3643086083981, 110557605978901, 3579776914324250, 123074955978249433, 4474133111905169219, 171363047274358839412, 6893620459732188296591, 290475101469031118494993

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Examples

			G.f. = 1 + 2*x + 13*x^2 + 143*x^3 + 1852*x^4 + 27563*x^5 + ...
		

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..180
• Moussa Benoumhani, On Whitney numbers of Dowling lattices, Discrete Math. 159 (1996), no. 1-3, 13-33.

Crossrefs

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).

Programs

Magma

m:=20; c:=9; R:=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

Maple

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

Mathematica

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 *)
Table[Sum[Binomial[n, k] * 9^k * BellB[k, 1/9], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 17 2020 *)

PARI

Vec(serlaplace(exp(z + (exp(9*z) - 1)/9))) \\ Michel Marcus, Nov 07 2014

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

Formula

E.g.f.: exp(x + (exp(9*x) - 1)/9).

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

a(n) = exp(-1/9) * Sum_{k>=0} (9*k + 1)^n / (9^k * k!). - Ilya Gutkovskiy, Apr 16 2020

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

Extensions

Name clarified by Muniru A Asiru, Feb 24 2019