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%I #21 Sep 08 2022 08:45:01
%S 0,0,2,15,54,141,306,588,1036,1710,2682,4037,5874,8307,11466,15498,
%T 20568,26860,34578,43947,55214,68649,84546,103224,125028,150330,
%U 179530,213057,251370,294959,344346,400086,462768,533016,611490,698887,795942,903429,1022162
%N Number of proper T_1-hypergraphs with 3 labeled nodes and n hyperedges.
%C 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
%C 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
%D 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)
%D V. Jovovic, G. Kilibarda, On enumeration of the class of all monotone Boolean functions, in preparation.
%H G. C. Greubel, <a href="/A056078/b056078.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F 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).
%F a(n+1) = ( n^4 +20*n^3 +35*n^2 -140*n +84 )*n/120.
%F From _Colin Barker_, Jul 11 2013: (Start)
%F a(n) = (-240+394*n-135*n^2-35*n^3+15*n^4+n^5)/120.
%F G.f.: x^3 *(x-2) *(2*x^2-2*x-1) / (x-1)^6. (End)
%e 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}}.
%t Table[(n^4 + 20*n^3 + 35*n^2 - 140*n + 84)*n/120, {n, 0, 50}] (* _G. C. Greubel_, Oct 07 2017 *)
%o (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
%o (Magma) [(n^4 + 20*n^3 + 35*n^2 - 140*n + 84)*n/120: n in [0..25]]; // _G. C. Greubel_, Oct 07 2017
%Y Cf. A056005, A047707.
%K nonn,easy
%O 1,3
%A _Vladeta Jovovic_, Goran Kilibarda, Jul 26 2000
%E More terms from _Colin Barker_, Jul 11 2013
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