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%I #56 Jul 31 2022 20:53:51
%S 1,1,2,9,56,480,5094,65534,984808,16992144,330667680,7170714672,
%T 171438170232,4480972742064,127115833240200,3889913061111240,
%U 127729720697035584,4479821940873927168,167143865005981109952,6610411989494027218368,276242547290322145178880
%N Coefficients of iterated exponential function defined by y(x) = x^y(x) for e^-e < x < e^(1/e), expanded about x=1.
%C a(n) is the n-th derivative of x^(x^...(x^(x^x))) with n x's evaluated at x=1. - _Alois P. Heinz_, Oct 14 2016
%H Alois P. Heinz, <a href="/A033917/b033917.txt">Table of n, a(n) for n = 0..400</a>
%H R. Arthur Knoebel, <a href="https://www.maa.org/programs/maa-awards/writing-awards/exponentials-reiterated-0">Exponentials Reiterated</a>, Amer. Math. Monthly 88 (1981), pp. 235-252.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PowerTower.html">Power Tower</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation">Knuth's up-arrow notation</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetration">Tetration</a>
%F E.g.f.: -LambertW(-log(1+x))/log(1+x). a(n) = Sum_{k=0..n} Stirling1(n, k)*(k+1)^(k-1). - _Vladeta Jovovic_, Nov 12 2003
%F a(n) ~ n^(n-1) / ( exp(n -3/2 + exp(-1)/2) * (exp(exp(-1))-1)^(n-1/2) ). - _Vaclav Kotesovec_, Nov 27 2012
%F E.g.f.: A(x) satisfies A(x) = Sum_{n>=0} x^n/n! * Sum_{k=0..n} Stirling1(n,k) * A(x)^k. - _Paul D. Hanna_, Mar 09 2013
%F a(n) = n! * [x^n] (x+1)^^n. - _Alois P. Heinz_, Oct 19 2016
%p a:= n-> add(Stirling1(n, k)*(k+1)^(k-1), k=0..n):
%p seq(a(n), n=0..25); # _Alois P. Heinz_, Aug 31 2012
%t mx = 20; Table[ i! SeriesCoefficient[ InverseSeries[ Series[ y^(1/y), {y, 1, mx}]], i], {i, 0, n}] (* modified by _Robert G. Wilson v_, Feb 03 2013 *)
%t CoefficientList[Series[-LambertW[-Log[1+x]]/Log[1+x], {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Nov 27 2012 *)
%o (PARI) Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)
%o a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, sum(k=0, m, Stirling1(m, k)*(A+x*O(x^n))^k)*x^m/m!)); n!*polcoeff(A, n)
%o for(n=0,20,print1(a(n),", ")) \\ _Paul D. Hanna_, Mar 09 2013
%Y Cf. A052880, A215703, A216349, A216350.
%Y Row sums of A277536.
%Y Main diagonal of A277537.
%K nonn
%O 0,3
%A _Patrick D McLean_
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