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A003578
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Dowling numbers: e.g.f. exp(x + (exp(b*x) - 1)/b), with b=6.
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15
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1, 2, 10, 80, 772, 8648, 111592, 1631360, 26518672, 472528160, 9139219360, 190461416192, 4250569655872, 101040920561792, 2546488866632320, 67772341398044672, 1898177372174512384, 55780954727160472064, 1715291443214323558912, 55062161002484359565312
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OFFSET
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0,2
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LINKS
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FORMULA
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E.g.f.: exp(x + (exp(6*x) - 1)/6).
a(n) = exp(-1/6) * Sum_{k>=0} (6*k + 1)^n / (6^k * k!). - Ilya Gutkovskiy, Apr 16 2020
a(n) ~ 6^(n + 1/6) * n^(n + 1/6) * exp(n/LambertW(6*n) - n - 1/6) / (sqrt(1 + LambertW(6*n)) * LambertW(6*n)^(n + 1/6)). - Vaclav Kotesovec, Jun 26 2022
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MAPLE
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seq(coeff(series(factorial(n)*exp(z+(1/6)*exp(6*z)-(1/6)), z, n+1), z, n), n = 0 .. 20); # Muniru A Asiru, Feb 23 2019
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MATHEMATICA
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With[{nn=20}, CoefficientList[Series[Exp[x+Exp[6x]/6-1/6], {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Jun 12 2017 *)
Table[Sum[Binomial[n, k] * 6^k * BellB[k, 1/6], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 17 2020 *)
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PROG
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(PARI) my(x='x+O('x^20)); b=6; Vec(serlaplace(exp(x +(exp(b*x)-1)/b))) \\ G. C. Greubel, Feb 24 2019
(Magma) m:=20; c:=6; 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=6; 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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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