A246472 Number of order-preserving (monotone) functions from the power set of 1 = {0} to the power set of n = {0, ..., n-1}.

1, 3, 9, 30, 109, 418, 1650, 6604, 26589, 107274, 432934, 1746484, 7040626, 28362324, 114175812, 459344920, 1847008989, 7423262554, 29822432862, 119766845860, 480833598054, 1929896415484, 7744047734652, 31067665113640, 124613703290994, 499744683756868

Offset: 0

Author

Jesse Han, Aug 27 2014

Keywords

• nonn

Comments

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.

Crossrefs

Matches A129167 with offset 2 for the first four terms.

Programs

Mathematica

Sum[Binomial[#,i](1+ Sum[Binomial[#,j],{j,i+1,#}]),{i,0,#}]& /@ Range[0,20]

PARI

a(n) = sum(i=0, n, binomial(n,i)*(1+ sum(j = i+1, n, binomial(n,j)))); \\ Michel Marcus, Aug 27 2014

Formula

a(n) = Sum_{i=0..n} (binomial(n,i)*(1 + Sum_{j=i+1..n} binomial(n,j))).

a(n) = 2^(2*n-1) + 2^n - binomial(2*n, n)/2. - Vaclav Kotesovec, Aug 28 2014

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