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

1, 2, 67, 4355, 295234, 21036803, 1625419909, 140823067772, 13947448935109, 1570142163116087, 196457384808738412, 26717651072732512841, 3896182904620308595021, 605803757139146097600266, 100236348400243756326661039, 17619174544126256877550593743, 3280792242500933388439611444802

Offset: 0

Author

Stefano Spezia, Jul 04 2023

Keywords

• nonn

Comments

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.

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), A364070 (b=624).

Row sums of the triangle A364072.

2nd row of the array A364074.

Programs

Mathematica

With[{m=16, b=63}, 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]63^(n-d-k),{d,0,n-k}],{k,0,n}]; Array[a,17,0]

Formula

E.g.f.: exp(x + (exp(63*x) - 1)/63).

a(n) = exp(-1/63) * Sum_{k>=0} (63*k + 1)^n / (63^k * k!).

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)).

a(n) = Sum_{k=0..n} Sum_{d=0..n-k} binomial(n, d)*StirlingS2(n-d, k)*63^(n-d-k).