A059585 - OEIS

%I #22 Sep 08 2022 08:45:03

%S 0,0,12,68,235,636,1478,3088,5958,10800,18612,30756,49049,75868,

%T 114270,168128,242284,342720,476748,653220,882759,1178012,1553926,

%U 2028048,2620850,3356080,4261140,5367492,6711093,8332860,10279166,12602368

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

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

%H G. C. Greubel, <a href="/A059585/b059585.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8,-1).

%F 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.

%F G.f.: x^2*(2-x)^2*(3-4*x+2*x^2)/(1-x)^8. - _Colin Barker_, Jun 25 2012

%p 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:

%t 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 *)

%t LinearRecurrence[{8, -28, 56, -70, 56, -28, 8, -1}, {0, 0, 12, 68, 235, 636, 1478, 3088}, 33] (* _Vincenzo Librandi_, Oct 07 2017 *)

%o (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

%o (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

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

%K easy,nonn

%O 0,3

%A _Vladeta Jovovic_, Goran Kilibarda, Jan 23 2001

%E More terms from _James A. Sellers_, Jan 24 2001