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

1, 2, 628, 393128, 247268752, 156500388128, 100264147266880, 65739252669562496, 44949841635462426880, 32961816599696140935680, 26763226019573589904012288, 24577197816669853786615064576, 25455086256328481246829666144256, 29063231104986184254344094194278400

Offset: 0

Author

Stefano Spezia, Jul 04 2023

Keywords

• nonn

Comments

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.

Links

• Prudence Djagba and Jan Hązła, Combinatorics of subgroups of Beidleman near-vector spaces, arXiv:2306.16421 [math.RA], 2023. See pp. 7-8.
• Paweł Hitczenko, A class of polynomial recurrences resulting in (n/log n, n/log^2 n)-asymptotic normality, arXiv:2403.03422 [math.CO], 2024. See p. 8.

Crossrefs

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).

Row sums of the triangle A364073.

3rd row of the array A364074.

Programs

Mathematica

With[{m=13, b=624}, CoefficientList[Series[Exp[x +(Exp[b*x]-1)/b], {x, 0, m}], x]*Range[0, m]!] (* or *)
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]

Formula

E.g.f.: exp(x + (exp(624*x) - 1)/624).

a(n) = exp(-1/624) * Sum_{k>=0} (624*k + 1)^n / (624^k * k!).

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)).

a(n) = Sum_{k=0..n} Sum_{d=0..n-k} binomial(n, d)*StirlingS2(n-d, k)*624^(n-d-k).