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A318180 Expansion of e.g.f. exp((1 - exp(-5*x))/5).

%I A318180 #19 Feb 16 2025 08:33:56
%S A318180 1,1,-4,11,21,-674,6551,-33479,-174114,7478121,-117699599,1090997976,
%T A318180 865365421,-302755297739,7922094623596,-127940743443649,
%U A318180 974028543402401,21377262410290446,-1179125036786673989,31760741865879345821,-552216474702144564074,2814873629049018241701
%N A318180 Expansion of e.g.f. exp((1 - exp(-5*x))/5).
%H A318180 Robert Israel, <a href="/A318180/b318180.txt">Table of n, a(n) for n = 0..459</a>
%H A318180 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>
%F A318180 a(n) = Sum_{k=0..n} (-5)^(n-k)*Stirling2(n,k).
%F A318180 a(0) = 1; a(n) = Sum_{k=1..n} (-5)^(k-1)*binomial(n-1,k-1)*a(n-k).
%F A318180 a(n) = (-5)^n*BellPolynomial_n(-1/5). - _Peter Luschny_, Aug 20 2018
%p A318180 seq((-5)^n*BellB(n,-1/5),n=0..30); # _Robert Israel_, Nov 11 2020
%t A318180 nmax = 21; CoefficientList[Series[Exp[(1 - Exp[-5 x])/5], {x, 0, nmax}], x] Range[0, nmax]!
%t A318180 Table[Sum[(-5)^(n - k) StirlingS2[n, k], {k, 0, n}], {n, 0, 21}]
%t A318180 a[n_] := a[n] = Sum[(-5)^(k - 1) Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 21}]
%t A318180 Table[(-5)^n BellB[n, -1/5], {n, 0, 21}] (* _Peter Luschny_, Aug 20 2018 *)
%o A318180 (PARI) my(x = 'x + O('x^25)); Vec(serlaplace(exp((1 - exp(-5*x))/5))) \\ _Michel Marcus_, Nov 11 2020
%Y A318180 Cf. A005011, A008548, A009235, A014182, A317996, A318179, A318181.
%K A318180 sign
%O A318180 0,3
%A A318180 _Ilya Gutkovskiy_, Aug 20 2018