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A033917 Coefficients of iterated exponential function defined by y(x) = x^y(x) for e^-e < x < e^(1/e), expanded about x=1. 26
1, 1, 2, 9, 56, 480, 5094, 65534, 984808, 16992144, 330667680, 7170714672, 171438170232, 4480972742064, 127115833240200, 3889913061111240, 127729720697035584, 4479821940873927168, 167143865005981109952, 6610411989494027218368, 276242547290322145178880 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
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
LINKS
R. Arthur Knoebel, Exponentials Reiterated, Amer. Math. Monthly 88 (1981), pp. 235-252.
Eric Weisstein's World of Mathematics, Power Tower
Wikipedia, Tetration
FORMULA
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
a(n) ~ n^(n-1) / ( exp(n -3/2 + exp(-1)/2) * (exp(exp(-1))-1)^(n-1/2) ). - Vaclav Kotesovec, Nov 27 2012
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
a(n) = n! * [x^n] (x+1)^^n. - Alois P. Heinz, Oct 19 2016
MAPLE
a:= n-> add(Stirling1(n, k)*(k+1)^(k-1), k=0..n):
seq(a(n), n=0..25); # Alois P. Heinz, Aug 31 2012
MATHEMATICA
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 *)
CoefficientList[Series[-LambertW[-Log[1+x]]/Log[1+x], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Nov 27 2012 *)
PROG
(PARI) Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)
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)
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Mar 09 2013
CROSSREFS
Row sums of A277536.
Main diagonal of A277537.
Sequence in context: A277539 A277540 A277541 * A277482 A274393 A004103
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified February 27 19:52 EST 2024. Contains 370378 sequences. (Running on oeis4.)