A003582 Dowling numbers: e.g.f. exp(x + (exp(b*x)-1)/b) with b=10.

1, 2, 14, 168, 2356, 37832, 701464, 14866848, 352943376, 9219925792, 261954304224, 8033968939648, 264411579439936, 9288709762556032, 346608927301622144, 13680000261825018368, 569006722158124974336, 24864267879086770135552, 1138321277772163220033024

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

In general, for b > 0, if e.g.f. = exp(x + (exp(b*x) - 1)/b), then a(n) ~ b^(n + 1/b) * n^(n + 1/b) * exp(n/LambertW(b*n) - n - 1/b) / (sqrt(1 + LambertW(b*n)) * LambertW(b*n)^(n + 1/b)). - Vaclav Kotesovec, Jun 26 2022

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.
• Paweł Hitczenko, A class of polynomial recurrences resulting in (n/log n, n/log^2 n)-asymptotic normality, arXiv:2403.03422 [math.CO], 2024. See p. 8.

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), A003581 (b=9), this sequence (b=10).

Programs

Magma

m:=20; c:=10; 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+(1/10)*exp(10*z)-(1/10)),z,n+1), z, n), n = 0 .. 20); # Muniru A Asiru, Feb 24 2019

Mathematica

With[{m=20, b=10}, 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] * 10^k * BellB[k, 1/10], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 17 2020 *)

PARI

my(x='x+O('x^20)); b=10; Vec(serlaplace(exp(x +(exp(b*x)-1)/b))) \\ G. C. Greubel, Feb 24 2019

Sage

m = 20; b=10; 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(10*x) - 1)/10).

a(n) = exp(-1/10) * Sum_{k>=0} (10*k + 1)^n / (10^k * k!). - Ilya Gutkovskiy, Apr 16 2020

a(n) ~ 10^(n + 1/10) * n^(n + 1/10) * exp(n/LambertW(10*n) - n - 1/10) / (sqrt(1 + LambertW(10*n)) * LambertW(10*n)^(n + 1/10)). - Vaclav Kotesovec, Jun 26 2022

Extensions

Name clarified by Muniru A Asiru, Feb 24 2019