A003580 Dowling numbers: e.g.f. exp(x + (exp(b*x)-1)/b) with b=8.

1, 2, 12, 120, 1424, 19488, 307904, 5539712, 111259904, 2454487552, 58847153152, 1522019629056, 42209521995776, 1248370355347456, 39186678731423744, 1300179383923212288, 45436201241711542272, 1667242078056889843712, 64063345344029286727680

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..190
• Moussa Benoumhani, On Whitney numbers of Dowling lattices, Discrete Math. 159 (1996), no. 1-3, 13-33.
• Prudence Djagba and Jan Hązła, Combinatorics of subgroups of Beidleman near-vector spaces, arXiv:2306.16421 [math.RA], 2023.

Crossrefs

Cf. A000110 (b=1), A007405 (b=2), A003575 (b=3), A003576 (b=4), A003577 (b=5), A003578 (b=6), A003579 (b=7), this sequence (b=8), A003581 (b=9), A003582 (b=10).

Programs

Magma

m:=20; c:=8; 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+(1/8)*exp(8*z)-(1/8)),z,n+1), z, n), n = 0 .. 20); # Muniru A Asiru, Feb 24 2019

Mathematica

With[{m=20, b=8}, 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] * 8^k * BellB[k, 1/8], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 17 2020 *)

PARI

my(x='x+O('x^20)); b=8; Vec(serlaplace(exp(x +(exp(b*x)-1)/b))) \\ G. C. Greubel, Feb 24 2019

Sage

m = 20; b=8; 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(8*x) - 1)/8).

a(n) = exp(-1/8) * Sum_{k>=0} (8*k + 1)^n / (8^k * k!). - Ilya Gutkovskiy, Apr 16 2020

a(n) ~ 8^(n + 1/8) * n^(n + 1/8) * exp(n/LambertW(8*n) - n - 1/8) / (sqrt(1 + LambertW(8*n)) * LambertW(8*n)^(n + 1/8)). - Vaclav Kotesovec, Jun 26 2022

Extensions

Name clarified by Muniru A Asiru, Feb 24 2019