A173403 - OEIS

%I #27 Jan 16 2024 04:11:06

%S 1,1,13,469,63577,33231721,68519123173,562469619451069,

%T 18442242396353040817,2417685638793025070212561,

%U 1267626422541873052658376446653,2658442047546208031914776455678477989,22300713297142388711251601783864453690641417

%N Inverse binomial transform of A002416.

%C 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.

%C 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

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

%H Brian Drake, <a href="/A173403/b173403.txt">Table of n, a(n) for n = 0..50</a>

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

%F a(n) ~ 2^(n^2). - _Vaclav Kotesovec_, Oct 30 2017

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

%t Table[Sum[(-1)^k Binomial[n,k] 2^(n-k)^2,{k,0,n}],{n,0,20}]  (* _Geoffrey Critzer_, Oct 02 2012 *)

%Y Cf. A002416, A135748, A287065, A048291.

%K nonn

%O 0,3

%A _Brian Drake_, Feb 17 2010