A003713 Expansion of e.g.f. log(1/(1+log(1-x))). (Formerly M1799 N0710)

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

Offset

0,3

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

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

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 *)

Prog

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

Crossrefs

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

Cf. A007840.

Cf. A238385.

Sequence in context: A185054 A014307 A000154 * A058129 A101514 A247240

Adjacent sequences: A003710 A003711 A003712 * A003714 A003715 A003716

Keyword

nonn,easy,nice

Author

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

Extensions

Thanks to Paul Zimmermann for comments.

Status

approved