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A364070 Dowling numbers: e.g.f. exp(x + (exp(b*x)-1)/b) with b=624.

%I A364070 #9 Mar 14 2024 15:56:56
%S A364070 1,2,628,393128,247268752,156500388128,100264147266880,
%T A364070 65739252669562496,44949841635462426880,32961816599696140935680,
%U A364070 26763226019573589904012288,24577197816669853786615064576,25455086256328481246829666144256,29063231104986184254344094194278400
%N A364070 Dowling numbers: e.g.f. exp(x + (exp(b*x)-1)/b) with b=624.
%C A364070 a(n) is the number of all 625-subgroups of R^n, where R^n is a near-vector space over a proper nearfield R.
%H A364070 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. See pp. 7-8.
%H A364070 Paweł Hitczenko, <a href="https://arxiv.org/abs/2403.03422">A class of polynomial recurrences resulting in (n/log n, n/log^2 n)-asymptotic normality</a>, arXiv:2403.03422 [math.CO], 2024. See p. 8.
%F A364070 E.g.f.: exp(x + (exp(624*x) - 1)/624).
%F A364070 a(n) = exp(-1/624) * Sum_{k>=0} (624*k + 1)^n / (624^k * k!).
%F A364070 a(n) ~ 624^(n + 1/624) * n^(n + 1/624) * exp(n/LambertW(624*n) - n - 1/624) / (sqrt(1 + LambertW(624*n)) * LambertW(624*n)^(n + 1/624)).
%F A364070 a(n) = Sum_{k=0..n} Sum_{d=0..n-k} binomial(n, d)*StirlingS2(n-d, k)*624^(n-d-k).
%t A364070 With[{m=13, b=624}, CoefficientList[Series[Exp[x +(Exp[b*x]-1)/b], {x, 0, m}], x]*Range[0, m]!] (* or *)
%t A364070 a[n_]:=Sum[Sum[Binomial[n,d]StirlingS2[n-d,k]624^(n-d-k),{d,0,n-k}],{k,0,n}]; Array[a,14,0]
%Y A364070 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), A003582 (b=10), A364069 (b=63).
%Y A364070 Row sums of the triangle A364073.
%Y A364070 3rd row of the array A364074.
%K A364070 nonn
%O A364070 0,2
%A A364070 _Stefano Spezia_, Jul 04 2023