A003713 Expansion of e.g.f. log(1/(1+log(1-x))).

0, 1, 2, 7, 35, 228, 1834, 17582, 195866, 2487832, 35499576, 562356672, 9794156448, 186025364016, 3826961710272, 84775065603888, 2011929826983504, 50929108873336320, 1369732445916318336, 39005083331889816960, 1172419218038422659456, 37095226237402478348544

Offset: 0

Author

N. J. A. Sloane, R. H. Hardin, Simon Plouffe

Keywords

• nonn
• easy
• nice

Comments

a(n+1) is the permanent of the n X n matrix M with M(i,i) = i+1, other entries 1. - Philippe Deléham, Nov 03 2005

Supernecklaces of type III (cycles of cycles). - Ricardo Bittencourt, May 05 2013

Unsigned coefficients for the raising / creation operator R for the Appell sequence of polynomials A238385: R = x + 1 - 2 D + 7 D^2/2! - 35 D^3/3! + ... . - Tom Copeland, May 09 2016

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 = 0..100
• P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
• Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 125.
• Jekuthiel Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353. [Annotated scanned copy]
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 34
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 298

Crossrefs

a(n)=|A039814(n, 1)| (first column of triangle). Cf. A000268, A000310, A000359, A000406, A001765.

Cf. A007840, A089064.

Cf. A238385.

Programs

Maple

series(ln(1/(1+ln(1-x))),x,17);
with (combstruct): M[ 1798 ] := [ A,{A=Cycle(Cycle(Z))},labeled ]:

Mathematica

With[{nn=20},CoefficientList[Series[Log[1/(1+Log[1-x])],{x,0,nn}],x]Range[0,nn]!] (* Harvey P. Dale, Dec 15 2012 *)
Table[Sum[(-1)^(n-k) * (k-1)! * StirlingS1[n, k], {k, 1, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 19 2024 *)

PARI

a(n)=if(n<0,0,n!*polcoeff(-log(1+log(1-x+x*O(x^n))),n))

Formula

Sum_{k=1..n} (k-1)!*|Stirling1(n, k)|. - Vladeta Jovovic, Sep 14 2003

a(n+1) = n! * Sum_{k=0..n} A007840(k)/k!. E.g., a(4) = 228 = 24*(1/1 + 1/1 + 3/2 + 14/6 + 88/24) = 24 + 24 + 36 + 56 + 88. - Philippe Deléham, Dec 10 2003

a(n) ~ (n-1)! * (exp(1)/(exp(1)-1))^n. - Vaclav Kotesovec, Jun 21 2013

a(0) = 0; a(n) = (n-1)! + Sum_{k=1..n-1} binomial(n-1,k) * (k-1)! * a(n-k). - Ilya Gutkovskiy, Jul 18 2020

Extensions

Thanks to Paul Zimmermann for comments.