A003579 Dowling numbers: e.g.f. exp(x + (exp(b*x) - 1)/b), with b=7.

1, 2, 11, 99, 1066, 13283, 190933, 3117900, 56729565, 1132679479, 24564972756, 574431351673, 14394977015245, 384489778509034, 10894501505088695, 326149933663962479, 10280153573323314858

Offset

0,2

Links

Muniru A Asiru, Table of n, a(n) for n = 0..425

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

Formula

E.g.f.: exp(x + (exp(7*x) - 1)/7).

a(n) = exp(-1/7) * Sum_{k>=0} (7*k + 1)^n / (7^k * k!). - Ilya Gutkovskiy, Apr 16 2020

a(n) ~ 7^(n + 1/7) * n^(n + 1/7) * exp(n/LambertW(7*n) - n - 1/7) / (sqrt(1 + LambertW(7*n)) * LambertW(7*n)^(n + 1/7)). - Vaclav Kotesovec, Jun 26 2022

Mathematica

With[{m=20, b=7}, 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] * 7^k * BellB[k, 1/7], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 17 2020 *)

Prog

(PARI) Vec(serlaplace( exp(z + 1/7 * exp(7 * z) - 1/7) ) ) \\ Joerg Arndt, Feb 24 2019
(Magma) m:=20; c:=7; R<x>:=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
(Sage) m = 20; b=7; 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
(GAP) b:=7;; 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

Crossrefs

Cf. A000110 (b=1), A007405 (b=2), A003575 (b=3), A003576 (b=4), A003577 (b=5), A003578 (b=6), this sequence (b=7), A003580 (b=8), A003581 (b=9), A003582 (b=10).

Sequence in context: A362467 A020559 A230889 * A282640 A099169 A143135

Adjacent sequences: A003576 A003577 A003578 * A003580 A003581 A003582

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Name clarified by Muniru A Asiru, Feb 24 2019

Status

approved