A059585 Number of labeled 3-node T_0-hypergraphs with n hyperedges (empty hyperedges and multiple hyperedges included).

0, 0, 12, 68, 235, 636, 1478, 3088, 5958, 10800, 18612, 30756, 49049, 75868, 114270, 168128, 242284, 342720, 476748, 653220, 882759, 1178012, 1553926, 2028048, 2620850, 3356080, 4261140, 5367492, 6711093, 8332860, 10279166, 12602368

Offset: 0

Author

Vladeta Jovovic, Goran Kilibarda, Jan 23 2001

Keywords

• easy
• nonn

Comments

A hypergraph is a T_0 hypergraph if for every two distinct nodes there exists a hyperedge containing one but not the other node.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Crossrefs

Cf. A059084, a(n)=A059584(n, 3), A059586.

Programs

Magma

[n*(n-1)*(n+1)*(n^4+28*n^3+323*n^2+1988*n+ 4572)/5040: n in [0..35]]; // Vincenzo Librandi, Oct 07 2017

Maple

for n from 0 to 100 do printf(`%d,`,n*(n - 1)*(n + 1)*(n^4 + 28*n^3 + 323*n^2 + 1988*n + 4572)/5040) od:

Mathematica

CoefficientList[Series[x^2*(2 - x)^2*(3 - 4*x + 2*x^2)/(1 - x)^8, {x, 0, 50}], x] (* G. C. Greubel, Oct 06 2017 *)
LinearRecurrence[{8, -28, 56, -70, 56, -28, 8, -1}, {0, 0, 12, 68, 235, 636, 1478, 3088}, 33] (* Vincenzo Librandi, Oct 07 2017 *)

PARI

x='x+O('x^50); concat([0,0],Vec(x^2*(2-x)^2*(3-4*x+2*x^2)/(1-x)^8)) \\ G. C. Greubel, Oct 06 2017

Formula

a(n) = binomial(n + 7, n) - 3*binomial(n + 3, n) + 2*binomial(n + 1, n) = n*(n - 1)*(n + 1)*(n^4 + 28*n^3 + 323*n^2 + 1988*n + 4572)/5040.

G.f.: x^2*(2-x)^2*(3-4*x+2*x^2)/(1-x)^8. - Colin Barker, Jun 25 2012

Extensions

More terms from James Sellers, Jan 24 2001