A000310 Coefficients of iterated exponentials.

1, 4, 26, 234, 2696, 37919, 630521, 12111114, 264051201, 6445170229, 174183891471, 5164718385337, 166737090160871, 5822980248613990, 218756388226681557, 8797723991458469015, 377159237609540937788, 17170729962232112834302, 827382365085791968518198, 42070004707327023844695198

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

References

• J. Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n=1..100
• P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
• Jekuthiel Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353. [Annotated scanned copy]
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 300

Crossrefs

a(n) = |A039816(n, 1)| (first column of triangle). Cf. A003713, A000268, A000359, A000406, A001765.

Programs

Mathematica

max = 20; CoefficientList[-Log[1 + Log[1 + Log[1 + Log[1 - x]]]]/x + O[x]^max, x]*Range[max]! (* Jean-François Alcover, Feb 07 2016 *)

PARI

T(n, k) = if(k==1, (n-1)!, sum(j=1, n, abs(stirling(n, j, 1))*T(j, k-1)));
a(n) = T(n, 4); \\ Seiichi Manyama, Feb 11 2022

PARI

my(x='x+O('x^40)); Vec(serlaplace(-log(1+log(1+log(1+log(1-x)))))); \\ Michel Marcus, Feb 11 2022

Formula

E.g.f.: -log(1+log(1+log(1+log(1-x)))).