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Expansion of e.g.f. log(1/(1+log(1-x))).
(Formerly M1799 N0710)
36

%I M1799 N0710 #76 Apr 19 2024 07:26:51

%S 0,1,2,7,35,228,1834,17582,195866,2487832,35499576,562356672,

%T 9794156448,186025364016,3826961710272,84775065603888,

%U 2011929826983504,50929108873336320,1369732445916318336,39005083331889816960,1172419218038422659456,37095226237402478348544

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

%C 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 Supernecklaces of type III (cycles of cycles). - _Ricardo Bittencourt_, May 05 2013

%C 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 J. Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A003713/b003713.txt">Table of n, a(n) for n = 0..100</a>

%H 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 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 Jekuthiel Ginsburg, <a href="/A000405/a000405.pdf">Iterated exponentials</a>, Scripta Math., 11 (1945), 340-353. [Annotated scanned copy]

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=34">Encyclopedia of Combinatorial Structures 34</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=298">Encyclopedia of Combinatorial Structures 298</a>

%F Sum_{k=1..n} (k-1)!*|Stirling1(n, k)|. - _Vladeta Jovovic_, Sep 14 2003

%F 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 a(n) ~ (n-1)! * (exp(1)/(exp(1)-1))^n. - _Vaclav Kotesovec_, Jun 21 2013

%F 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 series(ln(1/(1+ln(1-x))),x,17);

%p with (combstruct): M[ 1798 ] := [ A,{A=Cycle(Cycle(Z))},labeled ]:

%t 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 Table[Sum[(-1)^(n-k) * (k-1)! * StirlingS1[n, k], {k, 1, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Apr 19 2024 *)

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

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

%Y Cf. A007840, A089064.

%Y Cf. A238385.

%K nonn,easy,nice

%O 0,3

%A _N. J. A. Sloane_, _R. H. Hardin_, _Simon Plouffe_

%E Thanks to _Paul Zimmermann_ for comments.