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Number of labeled pure 2-complexes on n nodes (0-simplexes) with 5 2-simplexes and 10 1-simplexes.
8

%I #19 Sep 08 2022 08:45:01

%S 72,4824,32256,127008,378000,940464,2062368,4115232,7629336,13333320,

%T 22198176,35485632,54800928,82149984,120000960,171350208,239792616,

%U 329596344,445781952,594205920,781648560,1015906320,1305888480

%N Number of labeled pure 2-complexes on n nodes (0-simplexes) with 5 2-simplexes and 10 1-simplexes.

%C Number of {T_1,T_2,...,T_k} where T_i,i=1..k are 3-subsets of an n-set such that {D | D is 2-subset of T_i for some i=1..k} has l elements; k=5,l=10.

%D V. Jovovic, On the number of two-dimensional simplicial complexes (in Russian), Metody i sistemy tekhnicheskoy diagnostiki, Vypusk 16, Mezhvuzovskiy zbornik nauchnykh trudov, Izdatelstvo Saratovskogo universiteta, 1991.

%H Vincenzo Librandi, <a href="/A054557/b054557.txt">Table of n, a(n) for n = 5..1000</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).

%F a(n) = 72*C(n, 5)+4392*C(n, 6) = n*(n-1)*(n-2)*(n-3)*(n-4)*(61*n-299)/10.

%F G.f.: 72*x^5*(1+60*x)/(1-x)^7. - Colin Barker, Jan 19 2012

%F a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7). _Vincenzo Librandi_, Apr 28 2012

%t CoefficientList[Series[72*(1+60*x)/(1-x)^7,{x,0,30}],x] (* _Vincenzo Librandi_, Apr 28 2012 *)

%o (Magma) I:=[72, 4824, 32256, 127008, 378000, 940464, 2062368]; [n le 7 select I[n] else 7*Self(n-1)-21*Self(n-2)+35*Self(n-3)-35*Self(n-4)+21*Self(n-5)-7*Self(n-6)+Self(n-7): n in [1..25]]; // _Vincenzo Librandi_, Apr 28 2012

%K nonn,easy

%O 5,1

%A _Vladeta Jovovic_, Apr 10 2000

%E More terms from _James A. Sellers_, Apr 11 2000