A003578 Dowling numbers: e.g.f. exp(x + (exp(b*x) - 1)/b), with b=6.

1, 2, 10, 80, 772, 8648, 111592, 1631360, 26518672, 472528160, 9139219360, 190461416192, 4250569655872, 101040920561792, 2546488866632320, 67772341398044672, 1898177372174512384, 55780954727160472064, 1715291443214323558912, 55062161002484359565312

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

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), this sequence (b=6), A003579 (b=7), A003580 (b=8), A003581 (b=9), A003582 (b=10).

Programs

Magma

m:=20; c:=6; 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+(1/6)*exp(6*z)-(1/6)),z,n+1), z, n), n = 0 .. 20); # Muniru A Asiru, Feb 23 2019

Mathematica

With[{nn=20},CoefficientList[Series[Exp[x+Exp[6x]/6-1/6],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 12 2017 *)
Table[Sum[Binomial[n, k] * 6^k * BellB[k, 1/6], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 17 2020 *)

PARI

my(x='x+O('x^20)); b=6; Vec(serlaplace(exp(x +(exp(b*x)-1)/b))) \\ G. C. Greubel, Feb 24 2019

Sage

m = 20; b=6; 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(6*x) - 1)/6).

a(n) = exp(-1/6) * Sum_{k>=0} (6*k + 1)^n / (6^k * k!). - Ilya Gutkovskiy, Apr 16 2020

a(n) ~ 6^(n + 1/6) * n^(n + 1/6) * exp(n/LambertW(6*n) - n - 1/6) / (sqrt(1 + LambertW(6*n)) * LambertW(6*n)^(n + 1/6)). - Vaclav Kotesovec, Jun 26 2022