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A364069 Dowling numbers: e.g.f. exp(x + (exp(b*x)-1)/b) with b = 63.

%I A364069 #11 Mar 14 2024 15:57:00
%S A364069 1,2,67,4355,295234,21036803,1625419909,140823067772,13947448935109,
%T A364069 1570142163116087,196457384808738412,26717651072732512841,
%U A364069 3896182904620308595021,605803757139146097600266,100236348400243756326661039,17619174544126256877550593743,3280792242500933388439611444802
%N A364069 Dowling numbers: e.g.f. exp(x + (exp(b*x)-1)/b) with b = 63.
%C A364069 a(n) is the number of all 64-subgroups of R^n, where R^n is a near-vector space over a proper nearfield R.
%H A364069 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 A364069 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 A364069 E.g.f.: exp(x + (exp(63*x) - 1)/63).
%F A364069 a(n) = exp(-1/63) * Sum_{k>=0} (63*k + 1)^n / (63^k * k!).
%F A364069 a(n) ~ 63^(n + 1/63) * n^(n + 1/63) * exp(n/LambertW(63*n) - n - 1/63) / (sqrt(1 + LambertW(63*n)) * LambertW(63*n)^(n + 1/63)).
%F A364069 a(n) = Sum_{k=0..n} Sum_{d=0..n-k} binomial(n, d)*StirlingS2(n-d, k)*63^(n-d-k).
%t A364069 With[{m=16, b=63}, CoefficientList[Series[Exp[x +(Exp[b*x]-1)/b], {x, 0, m}], x]*Range[0, m]!] (* or *)
%t A364069 a[n_]:=Sum[Sum[Binomial[n,d]StirlingS2[n-d,k]63^(n-d-k),{d,0,n-k}],{k,0,n}]; Array[a,17,0]
%Y A364069 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), A364070 (b=624).
%Y A364069 Row sums of the triangle A364072.
%Y A364069 2nd row of the array A364074.
%K A364069 nonn
%O A364069 0,2
%A A364069 _Stefano Spezia_, Jul 04 2023