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%I #27 Jul 06 2023 01:55:43
%S 1,2,12,120,1424,19488,307904,5539712,111259904,2454487552,
%T 58847153152,1522019629056,42209521995776,1248370355347456,
%U 39186678731423744,1300179383923212288,45436201241711542272,1667242078056889843712,64063345344029286727680
%N Dowling numbers: e.g.f. exp(x + (exp(b*x)-1)/b) with b=8.
%H Muniru A Asiru, <a href="/A003580/b003580.txt">Table of n, a(n) for n = 0..190</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.
%H Prudence Djagba and Jan Hązła, <a href="https://arxiv.org/abs/2306.16421">Combinatorics of subgroups of Beidleman near-vector spaces</a>, arXiv:2306.16421 [math.RA], 2023.
%F E.g.f.: exp(x + (exp(8*x) - 1)/8).
%F a(n) = exp(-1/8) * Sum_{k>=0} (8*k + 1)^n / (8^k * k!). - _Ilya Gutkovskiy_, Apr 16 2020
%F a(n) ~ 8^(n + 1/8) * n^(n + 1/8) * exp(n/LambertW(8*n) - n - 1/8) / (sqrt(1 + LambertW(8*n)) * LambertW(8*n)^(n + 1/8)). - _Vaclav Kotesovec_, Jun 26 2022
%p seq(coeff(series(factorial(n)*exp(z+(1/8)*exp(8*z)-(1/8)),z,n+1), z, n), n = 0 .. 20); # _Muniru A Asiru_, Feb 24 2019
%t With[{m=20, b=8}, 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] * 8^k * BellB[k, 1/8], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Apr 17 2020 *)
%o (PARI) my(x='x+O('x^20)); b=8; Vec(serlaplace(exp(x +(exp(b*x)-1)/b))) \\ _G. C. Greubel_, Feb 24 2019
%o (Magma) m:=20; c:=8; 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=8; 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
%Y Cf. A000110 (b=1), A007405 (b=2), A003575 (b=3), A003576 (b=4), A003577 (b=5), A003578 (b=6), A003579 (b=7), this sequence (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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