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
a(n) ~ 5^(n + 1/5) * n^(n + 1/5) * exp(n/LambertW(5*n) - n - 1/5) / (sqrt(1 + LambertW(5*n)) * LambertW(5*n)^(n + 1/5)). - Vaclav Kotesovec, Jun 26 2022
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<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=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
KEYWORD
nonn
AUTHOR
EXTENSIONS
Name clarified by Muniru A Asiru, Feb 24 2019
STATUS
approved