A056078 Number of proper T_1-hypergraphs with 3 labeled nodes and n hyperedges.

0, 0, 2, 15, 54, 141, 306, 588, 1036, 1710, 2682, 4037, 5874, 8307, 11466, 15498, 20568, 26860, 34578, 43947, 55214, 68649, 84546, 103224, 125028, 150330, 179530, 213057, 251370, 294959, 344346, 400086, 462768, 533016, 611490, 698887, 795942, 903429, 1022162

Offset: 1

Author

Vladeta Jovovic, Goran Kilibarda, Jul 26 2000

Keywords

• nonn
• easy

Comments

Also number of 3 X 3 matrices with nonnegative integer entries with zero main diagonal and without zero rows or columns, such that sum of all entries is n. - Vladeta Jovovic, Sep 06 2006

A T_1-hypergraph is a hypergraph (not necessarily without empty hyperedges or multiple hyperedges) which for every ordered pair (u,v) of distinct nodes has a hyperedge containing u but not v. A proper hypergraph is a hypergraph without empty hyperedges or hyperedges containing all nodes. - Vladeta Jovovic, Sep 06 2006

Examples

			There are 15 proper T_1-hypergraphs with 3 nodes and 4 hyperedges: {{3},{3},{2},{1}}, {{3},{2},{2},{1}}, {{3},{2},{2,3},{1}}, {{3},{2},{1},{1}}, {{3},{2},{1},{1,3}}, {{3},{2},{1},{1,2}}, {{3},{2},{1,3},{1,2}}, {{3},{2,3},{1},{1,2}}, {{3},{2,3},{1,3},{1,2}}, {{2},{2,3},{1},{1,3}}, {{2},{2,3},{1,3},{1,2}}, {{2,3},{2,3},{1,3},{1,2}}, {{2,3},{1},{1,3},{1,2}}, {{2,3},{1,3},{1,3},{1,2}}, {{2,3},{1,3},{1,2},{1,2}}.
		

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 = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A056005, A047707.

Programs

Magma

[(n^4 + 20*n^3 + 35*n^2 - 140*n + 84)*n/120: n in [0..25]]; // G. C. Greubel, Oct 07 2017

Mathematica

Table[(n^4 + 20*n^3 + 35*n^2 - 140*n + 84)*n/120, {n, 0, 50}] (* G. C. Greubel, Oct 07 2017 *)

PARI

for(n=0,25, print1((n^4 + 20*n^3 + 35*n^2 - 140*n + 84)*n/120, ", ")) \\ G. C. Greubel, Oct 07 2017

Formula

a(n) = C(n+5,5) -6*C(n+3,3) +6*C(n+2,2) +3*C(n+1,1) -6*C(n,0).

a(n+1) = ( n^4 +20*n^3 +35*n^2 -140*n +84 )*n/120.

From Colin Barker, Jul 11 2013: (Start)

a(n) = (-240+394*n-135*n^2-35*n^3+15*n^4+n^5)/120.

G.f.: x^3 *(x-2) *(2*x^2-2*x-1) / (x-1)^6. (End)

Extensions

More terms from Colin Barker, Jul 11 2013