A005327 Certain subgraphs of a directed graph (inverse binomial transform of A005321). (Formerly M4289)

1, 0, 1, 6, 91, 2820, 177661, 22562946, 5753551231, 2940064679040, 3007686166657921, 6156733583148764286, 25211824022994189751171, 206510050572345408251841660, 3383254158526734823389921915781

Offset

1,4

Comments

q-Subfactorial for q=2. - Vladimir Reshetnikov, Sep 12 2016

References

T. L. Greenough, Enumeration of interval orders without duplicated holdings, Preprint, circa 1976.

T. L. Greenough and T. Lockman, Representation and enumeration of interval orders and semiorders, Ph.D. Thesis, Dartmouth, 1976.

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

Links

G. C. Greubel, Table of n, a(n) for n = 1..50

E. Andresen and K. Kjeldsen, On certain subgraphs of a complete transitively directed graph, Discrete Math. 14 (1976), no. 2, 103-119.

T. L. Greenough, Enumeration of interval orders without duplicated holdings, Preprint, circa 1976. [Annotated scanned copy]

Hsien-Kuei Hwang, Emma Yu Jin, and Michael J. Schlosser, Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow, arXiv:2012.13570 [math.CO], 2020.

N. J. A. Sloane, Transforms

Eric Weisstein's World of Mathematics, Subfactorial, q-Factorial, q-Analog.

Formula

For n>1, a(n) = (2^(n-1)-1)*a(n-1) + (-1)^(n-1). - Max Alekseyev, Feb 23 2012

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

a(n) ~ A048651^2 * 2^(n*(n-1)/2). - Vaclav Kotesovec, Oct 09 2019

Maple

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

Mathematica

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 *)
With[{q = 2}, Table[QFactorial[n, q] Sum[(-1)^k/QFactorial[k, q], {k, 0, n}], {n, 0, 20}]] (* Vladimir Reshetnikov, Sep 12 2016 *)

Crossrefs

Cf. A002820, A005329, A005321.

Sequence in context: A349716 A219220 A006151 * A182263 A360826 A171910

Adjacent sequences: A005324 A005325 A005326 * A005328 A005329 A005330

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Max Alekseyev, Apr 27 2010

Status

approved