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

%I A003713 M1799 N0710 #76 Apr 19 2024 07:26:51
%S A003713 0,1,2,7,35,228,1834,17582,195866,2487832,35499576,562356672,
%T A003713 9794156448,186025364016,3826961710272,84775065603888,
%U A003713 2011929826983504,50929108873336320,1369732445916318336,39005083331889816960,1172419218038422659456,37095226237402478348544
%N A003713 Expansion of e.g.f. log(1/(1+log(1-x))).
%C A003713 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
%C A003713 Supernecklaces of type III (cycles of cycles). - _Ricardo Bittencourt_, May 05 2013
%C A003713 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
%D A003713 J. Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353.
%D A003713 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A003713 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003713 T. D. Noe, <a href="/A003713/b003713.txt">Table of n, a(n) for n = 0..100</a>
%H A003713 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A003713 Philippe Flajolet and Robert Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/AnaCombi/anacombi.html">Analytic Combinatorics</a>, Cambridge Univ. Press, 2009, page 125.
%H A003713 Jekuthiel Ginsburg, <a href="/A000405/a000405.pdf">Iterated exponentials</a>, Scripta Math., 11 (1945), 340-353. [Annotated scanned copy]
%H A003713 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=34">Encyclopedia of Combinatorial Structures 34</a>
%H A003713 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=298">Encyclopedia of Combinatorial Structures 298</a>
%F A003713 Sum_{k=1..n} (k-1)!*|Stirling1(n, k)|. - _Vladeta Jovovic_, Sep 14 2003
%F A003713 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
%F A003713 a(n) ~ (n-1)! * (exp(1)/(exp(1)-1))^n. - _Vaclav Kotesovec_, Jun 21 2013
%F A003713 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
%p A003713 series(ln(1/(1+ln(1-x))),x,17);
%p A003713 with (combstruct): M[ 1798 ] := [ A,{A=Cycle(Cycle(Z))},labeled ]:
%t A003713 With[{nn=20},CoefficientList[Series[Log[1/(1+Log[1-x])],{x,0,nn}],x]Range[0,nn]!] (* _Harvey P. Dale_, Dec 15 2012 *)
%t A003713 Table[Sum[(-1)^(n-k) * (k-1)! * StirlingS1[n, k], {k, 1, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Apr 19 2024 *)
%o A003713 (PARI) a(n)=if(n<0,0,n!*polcoeff(-log(1+log(1-x+x*O(x^n))),n))
%Y A003713 a(n)=|A039814(n, 1)| (first column of triangle). Cf. A000268, A000310, A000359, A000406, A001765.
%Y A003713 Cf. A007840, A089064.
%Y A003713 Cf. A238385.
%K A003713 nonn,easy,nice
%O A003713 0,3
%A A003713 _N. J. A. Sloane_, _R. H. Hardin_, _Simon Plouffe_
%E A003713 Thanks to _Paul Zimmermann_ for comments.