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%I #28 Jul 15 2017 10:36:17
%S 1,3,9,30,109,418,1650,6604,26589,107274,432934,1746484,7040626,
%T 28362324,114175812,459344920,1847008989,7423262554,29822432862,
%U 119766845860,480833598054,1929896415484,7744047734652,31067665113640,124613703290994,499744683756868
%N Number of order-preserving (monotone) functions from the power set of 1 = {0} to the power set of n = {0, ..., n-1}.
%C This is the number of ways to choose a pair of elements (x,y) of P(n) so that x is a subset of y. This also gives the number of covariant functors from P(1) to P(n) viewed as categories.
%F a(n) = sum_{i=0..n} (binomial(n,i)*(1 + sum_{j=i+1..n} binomial(n,j)).
%F a(n) = 2^(2*n-1) + 2^n - binomial(2*n, n)/2. - _Vaclav Kotesovec_, Aug 28 2014
%F n*(n-4)*a(n) +2*(-5*n^2+23*n-15)*a(n-1) +4*(8*n^2-41*n+45)*a(n-2) -16*(2*n-5)*(n-3)*a(n-3)=0. - _R. J. Mathar_, Jul 15 2017
%t Sum[Binomial[#,i](1+ Sum[Binomial[#,j],{j,i+1,#}]),{i,0,#}]& /@ Range[0,20]
%o (PARI) a(n) = sum(i=0, n, binomial(n,i)*(1+ sum(j = i+1, n, binomial(n,j)))); \\ _Michel Marcus_, Aug 27 2014
%Y Matches A129167 with offset 2 for the first four terms.
%K nonn
%O 0,2
%A _Jesse Han_, Aug 27 2014
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