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%I M4289 #49 Apr 13 2021 17:30:52
%S 1,0,1,6,91,2820,177661,22562946,5753551231,2940064679040,
%T 3007686166657921,6156733583148764286,25211824022994189751171,
%U 206510050572345408251841660,3383254158526734823389921915781
%N Certain subgraphs of a directed graph (inverse binomial transform of A005321).
%C q-Subfactorial for q=2. - _Vladimir Reshetnikov_, Sep 12 2016
%D T. L. Greenough, Enumeration of interval orders without duplicated holdings, Preprint, circa 1976.
%D T. L. Greenough and T. Lockman, Representation and enumeration of interval orders and semiorders, Ph.D. Thesis, Dartmouth, 1976.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H G. C. Greubel, <a href="/A005327/b005327.txt">Table of n, a(n) for n = 1..50</a>
%H E. Andresen and K. Kjeldsen, <a href="http://dx.doi.org/10.1016/0012-365X(76)90054-6">On certain subgraphs of a complete transitively directed graph</a>, Discrete Math. 14 (1976), no. 2, 103-119.
%H T. L. Greenough, <a href="/A005321/a005321_1.pdf">Enumeration of interval orders without duplicated holdings</a>, Preprint, circa 1976. [Annotated scanned copy]
%H Hsien-Kuei Hwang, Emma Yu Jin, and Michael J. Schlosser, <a href="https://arxiv.org/abs/2012.13570">Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow</a>, arXiv:2012.13570 [math.CO], 2020.
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Subfactorial.html">Subfactorial</a>, <a href="http://mathworld.wolfram.com/q-Factorial.html">q-Factorial</a>, <a href="http://mathworld.wolfram.com/q-Analog.html">q-Analog</a>.
%F For n>1, a(n) = (2^(n-1)-1)*a(n-1) + (-1)^(n-1). - _Max Alekseyev_, Feb 23 2012
%F a(n) = p(n-1)*sum((-1)^j/p(j), j=0..n-1), where p(0) = 1, p(k) = product(2^i-1, i=1..k) for k>0. - _Emeric Deutsch_, Jan 23 2005
%F a(n) ~ A048651^2 * 2^(n*(n-1)/2). - _Vaclav Kotesovec_, Oct 09 2019
%p p:=proc(n) if n=0 then 1 else product(2^i-1,i=1..n) fi end: a:=n->p(n-1)*sum((-1)^j/p(j),j=0..n-1): seq(a(n),n=1..17); # _Emeric Deutsch_, Jan 23 2005
%t a[1] = 1; a[n_] := a[n] = (2^(n-1)-1)*a[n-1] + (-1)^(n-1); Array[a, 15] (* _Jean-François Alcover_, Apr 05 2016, after _Max Alekseyev_ *)
%t With[{q = 2}, Table[QFactorial[n, q] Sum[(-1)^k/QFactorial[k, q], {k, 0, n}], {n, 0, 20}]] (* _Vladimir Reshetnikov_, Sep 12 2016 *)
%Y Cf. A002820, A005329, A005321.
%K nonn
%O 1,4
%A _N. J. A. Sloane_
%E More terms from _Max Alekseyev_, Apr 27 2010
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