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A003575 Dowling numbers: e.g.f.: exp(x + (exp(b*x) - 1)/b) with b=3. 13
1, 2, 7, 35, 214, 1523, 12349, 112052, 1120849, 12219767, 143942992, 1819256321, 24526654381, 350974470746, 5308470041299, 84554039118383, 1413794176669942, 24745966692370607, 452277149756692105, 8612255652371171012, 170517319084490074405 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Muniru A Asiru, Table of n, a(n) for n = 0..210

Moussa Benoumhani, On Whitney numbers of Dowling lattices, Discrete Math. 159 (1996), no. 1-3, 13-33.

M. M. Mangontarum, J. Katriel, On q-Boson Operators and q-Analogues of the r-Whitney and r-Dowling Numbers, J. Int. Seq. 18 (2015) 15.9.8.

FORMULA

E.g.f.: exp(x + (exp(3*x) - 1)/3).

G.f.: 1/(1-x*Q(0)), where Q(k)= 1 + x/(1 - x + 3*x*(k+1)/(x - 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 19 2013

a(n) = exp(-1/3) * Sum_{k>=0} (3*k + 1)^n / (3^k * k!). - Ilya Gutkovskiy, Apr 16 2020

MAPLE

seq(coeff(series(n!*exp(z+(1/3)*exp(3*z)-(1/3)), z, n+1), z, n), n=0..30); # Muniru A Asiru, Feb 19 2019

MATHEMATICA

With[{nn=20}, CoefficientList[Series[Exp[x+Exp[3x]/3-1/3], {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Jan 04 2019 *)

Table[Sum[Binomial[n, k] * 3^k * BellB[k, 1/3], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 17 2020 *)

PROG

(PARI) x = 'x + O('x^30) ; Vec(serlaplace(exp(x + exp(3*x)/3 - 1/3))) \\ Michel Marcus, Feb 09 2018

(MAGMA) m:=30; c:=3; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x+(Exp(c*x)-1)/c) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Feb 20 2019

(Sage)

b=3;

def A003575_list(prec):

    P.<x> = PowerSeriesRing(QQ, prec)

    return P( exp(x +(exp(b*x)-1)/b) ).egf_to_ogf().list()

A003575_list(30) # G. C. Greubel, Feb 20 2019

CROSSREFS

Cf. A000110 (b=1), A007405 (b=2), this sequence (b=3), A003576 (b=4), A003577 (b=5), A003578 (b=6), A003579 (b=7), A003580 (b=8), A003581 (b=9), A003582 (b=10).

Sequence in context: A172511 A214461 A130458 * A043546 A307441 A260530

Adjacent sequences:  A003572 A003573 A003574 * A003576 A003577 A003578

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

Name clarified by G. C. Greubel, Feb 20 2019

STATUS

approved

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Last modified November 29 19:26 EST 2020. Contains 338769 sequences. (Running on oeis4.)