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 A003577 Dowling numbers: e.g.f. exp(x + (exp(b*x)-1)/b) with b=5. 11
 1, 2, 9, 63, 536, 5307, 60389, 775988, 11062391, 172638727, 2921519374, 53221709973, 1037320865141, 21517178350762, 472862758184789, 10966587174511443, 267502464814857936, 6842498829509972687, 183057455239626138009, 5110016898453125496548 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Muniru A Asiru, Table of n, a(n) for n = 0..440 Moussa Benoumhani, On Whitney numbers of Dowling lattices, Discrete Math. 159 (1996), no. 1-3, 13-33. FORMULA E.g.f.: exp(x + (exp(5*x) - 1)/5). a(n) = exp(-1/5) * Sum_{k>=0} (5*k + 1)^n / (5^k * k!). - Ilya Gutkovskiy, Apr 16 2020 MATHEMATICA With[{m=20, b=5}, 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] * 5^k * BellB[k, 1/5], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 17 2020 *) PROG (PARI) my(x='x+O('x^20)); b=5; Vec(serlaplace(exp(x +(exp(b*x)-1)/b))) \\ G. C. Greubel, Feb 24 2019 (MAGMA) m:=20; c:=5; R:=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=5; 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 (GAP) b:=5;; a:=[1, 2];; for n in [3..20] do a[n]:=2*a[n-1]+Sum([0..n-3], i->Binomial(n-2, i)*b^(n-2-i)*a[i+1]); od; Print(a); # Muniru A Asiru, Apr 10 2019 CROSSREFS Cf. A000110 (b=1), A007405 (b=2), A003575 (b=3), A003576 (b=4), this sequence (b=5), A003578 (b=6), A003579 (b=7), A003580 (b=8), A003581 (b=9), A003582 (b=10). Sequence in context: A100262 A166886 A212413 * A085928 A130169 A218672 Adjacent sequences:  A003574 A003575 A003576 * A003578 A003579 A003580 KEYWORD nonn AUTHOR EXTENSIONS Name clarified by Muniru A Asiru, Feb 24 2019 STATUS approved

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Last modified August 11 18:37 EDT 2020. Contains 336428 sequences. (Running on oeis4.)