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A003580
Dowling numbers: e.g.f. exp(x + (exp(b*x)-1)/b) with b=8.
15
1, 2, 12, 120, 1424, 19488, 307904, 5539712, 111259904, 2454487552, 58847153152, 1522019629056, 42209521995776, 1248370355347456, 39186678731423744, 1300179383923212288, 45436201241711542272, 1667242078056889843712, 64063345344029286727680
OFFSET
0,2
LINKS
Moussa Benoumhani, On Whitney numbers of Dowling lattices, Discrete Math. 159 (1996), no. 1-3, 13-33.
Prudence Djagba and Jan Hązła, Combinatorics of subgroups of Beidleman near-vector spaces, arXiv:2306.16421 [math.RA], 2023.
FORMULA
E.g.f.: exp(x + (exp(8*x) - 1)/8).
a(n) = exp(-1/8) * Sum_{k>=0} (8*k + 1)^n / (8^k * k!). - Ilya Gutkovskiy, Apr 16 2020
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
MAPLE
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
MATHEMATICA
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 *)
Table[Sum[Binomial[n, k] * 8^k * BellB[k, 1/8], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 17 2020 *)
PROG
(PARI) my(x='x+O('x^20)); b=8; Vec(serlaplace(exp(x +(exp(b*x)-1)/b))) \\ G. C. Greubel, Feb 24 2019
(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
(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
CROSSREFS
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).
Sequence in context: A329851 A127112 A365283 * A052580 A134095 A204042
KEYWORD
nonn
EXTENSIONS
Name clarified by Muniru A Asiru, Feb 24 2019
STATUS
approved