A056074 Number of 3-element ordered antichain covers of an unlabeled n-element set.
2, 17, 71, 212, 518, 1106, 2142, 3852, 6534, 10571, 16445, 24752, 36218, 51716, 72284, 99144, 133722, 177669, 232883, 301532, 386078, 489302, 614330, 764660, 944190, 1157247, 1408617, 1703576
Offset: 3
References
- V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6)
- V. Jovovic, G. Kilibarda, On enumeration of the class of all monotone Boolean functions, in preparation.
Links
- G. C. Greubel, Table of n, a(n) for n = 3..1000
- K. S. Brown, Dedekind's problem
- Eric Weisstein's World of Mathematics, Antichain covers
- Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).
Crossrefs
Programs
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Magma
[n*(n^5 + 21*n^4 - 5*n^3 - 345*n^2 + 724*n - 396)/720: n in [3..25]]; // G. C. Greubel, Oct 06 2017
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Maple
A056074:=n->n*(n^5 + 21*n^4 - 5*n^3 - 345*n^2 + 724*n - 396)/720: seq(A056074(n), n=3..60); # Wesley Ivan Hurt, Oct 06 2017
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Mathematica
LinearRecurrence[{7,-21,35,-35,21,-7,1},{2,17,71,212,518,1106,2142},30] (* or *) Table[Binomial[n+6,6]-6Binomial[n+4,4]+6Binomial[n+3,3]+ 3Binomial[n+2,2]- 6Binomial[n+1,1]+ 2Binomial[n,0],{n,3,30}] (* Harvey P. Dale, Jul 12 2011 *) -
PARI
a(n)=n*(n^5 + 21*n^4 - 5*n^3 - 345*n^2 + 724*n - 396)/720 \\ Charles R Greathouse IV, Feb 19 2017
Formula
a(n) = C(n + 6, 6) - 6*C(n + 4, 4) + 6*C(n + 3, 3) + 3*C(n + 2, 2) - 6*C(n + 1, 1) + 2*C(n, 0).
a(0)=2, a(1)=17, a(2)=71, a(3)=212, a(4)=518, a(5)=1106, a(6)=2142, a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7). - Harvey P. Dale, Jul 12 2011
G.f.: (-2-3*x+6*x^2-2*x^3)/(x-1)^7. - Harvey P. Dale, Jul 12 2011
a(n) = n*(n^5 + 21*n^4 - 5*n^3 - 345*n^2 + 724*n - 396)/720. - Charles R Greathouse IV, Feb 19 2017