A003577 Dowling numbers: e.g.f. exp(x + (exp(b*x)-1)/b) with b=5.

1, 2, 9, 63, 536, 5307, 60389, 775988, 11062391, 172638727, 2921519374, 53221709973, 1037320865141, 21517178350762, 472862758184789, 10966587174511443, 267502464814857936, 6842498829509972687, 183057455239626138009, 5110016898453125496548

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..440
• 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), this sequence (b=5), A003578 (b=6), A003579 (b=7), A003580 (b=8), A003581 (b=9), A003582 (b=10).

Programs

GAP

b:=5;; a:=[1,2];; for n in [3..20] do a[n]:=2*a[n-1]+Sum([0..n-3],i->Binomial(n-2,i)*b^(n-2-i)*a[i+1]); od; Print(a); # Muniru A Asiru, Apr 10 2019

Magma

m:=20; c:=5; 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

Mathematica

With[{m=20, b=5}, 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] * 5^k * BellB[k, 1/5], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 17 2020 *)

PARI

my(x='x+O('x^20)); b=5; Vec(serlaplace(exp(x +(exp(b*x)-1)/b))) \\ G. C. Greubel, Feb 24 2019

Sage

m = 20; b=5; 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(5*x) - 1)/5).

a(n) = exp(-1/5) * Sum_{k>=0} (5*k + 1)^n / (5^k * k!). - Ilya Gutkovskiy, Apr 16 2020

a(n) ~ 5^(n + 1/5) * n^(n + 1/5) * exp(n/LambertW(5*n) - n - 1/5) / (sqrt(1 + LambertW(5*n)) * LambertW(5*n)^(n + 1/5)). - Vaclav Kotesovec, Jun 26 2022

Extensions

Name clarified by Muniru A Asiru, Feb 24 2019