A173403 Inverse binomial transform of A002416.

1, 1, 13, 469, 63577, 33231721, 68519123173, 562469619451069, 18442242396353040817, 2417685638793025070212561, 1267626422541873052658376446653, 2658442047546208031914776455678477989, 22300713297142388711251601783864453690641417

Offset: 0

Author

Brian Drake, Feb 17 2010

Keywords

• nonn

Comments

a(n) is the number of n X n matrices of 0's and 1's with the property that there is no index k such that both the k-th column and the k-th row consist of all zeros.

a(n) is the number of binary relations on n labeled vertices with no vertex of indegree and outdegree = 0. - Geoffrey Critzer, Oct 02 2012

References

• E. A. Bender and S. G. Williamson, Foundations of Combinatorics with Applications, Dover, 2005, exercise 4.1.6.

Links

• Brian Drake, Table of n, a(n) for n = 0..50

Crossrefs

Cf. A002416, A135748, A287065, A048291.

Programs

Maple

N:=8: seq( sum(binomial(n,i)*2^((n-i)^2)*(-1)^(i), i=0..n), n=0..N);

Mathematica

Table[Sum[(-1)^k Binomial[n,k] 2^(n-k)^2,{k,0,n}],{n,0,20}]  (* Geoffrey Critzer, Oct 02 2012 *)

Formula

a(n) = Sum_{k=0..n} (-1)^k*binomial(n,k)*2^((n-k)^2).

a(n) ~ 2^(n^2). - Vaclav Kotesovec, Oct 30 2017