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

1,3

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

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).

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)

Example

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}}.

Mathematica

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

Prog

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

Crossrefs

Cf. A056005, A047707.

Sequence in context: A015520 A098520 A216333 * A142861 A305673 A268761

Adjacent sequences: A056075 A056076 A056077 * A056079 A056080 A056081

Keyword

nonn,easy

Author

Vladeta Jovovic, Goran Kilibarda, Jul 26 2000

Extensions

More terms from Colin Barker, Jul 11 2013

Status

approved