%I #32 Sep 08 2022 08:44:32
%S 1,2,11,99,1066,13283,190933,3117900,56729565,1132679479,24564972756,
%T 574431351673,14394977015245,384489778509034,10894501505088695,
%U 326149933663962479,10280153573323314858
%N Dowling numbers: e.g.f. exp(x + (exp(b*x) - 1)/b), with b=7.
%H Muniru A Asiru, <a href="/A003579/b003579.txt">Table of n, a(n) for n = 0..425</a>
%H Moussa Benoumhani, <a href="https://doi.org/10.1016/0012-365X(95)00095-E">On Whitney numbers of Dowling lattices</a>, Discrete Math. 159 (1996), no. 1-3, 13-33.
%F E.g.f.: exp(x + (exp(7*x) - 1)/7).
%F a(n) = exp(-1/7) * Sum_{k>=0} (7*k + 1)^n / (7^k * k!). - _Ilya Gutkovskiy_, Apr 16 2020
%F 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
%t 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 *)
%t Table[Sum[Binomial[n, k] * 7^k * BellB[k, 1/7], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Apr 17 2020 *)
%o (PARI) Vec(serlaplace( exp(z + 1/7 * exp(7 * z) - 1/7) ) ) \\ _Joerg Arndt_, Feb 24 2019
%o (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
%o (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
%o (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
%Y 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).
%K nonn
%O 0,2
%A _N. J. A. Sloane_
%E Name clarified by _Muniru A Asiru_, Feb 24 2019
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