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A054557 Number of labeled pure 2-complexes on n nodes (0-simplexes) with 5 2-simplexes and 10 1-simplexes.

%I A054557 #23 Jul 02 2025 16:01:59
%S A054557 72,4824,32256,127008,378000,940464,2062368,4115232,7629336,13333320,
%T A054557 22198176,35485632,54800928,82149984,120000960,171350208,239792616,
%U A054557 329596344,445781952,594205920,781648560,1015906320,1305888480,1661718240,2094838200,2618120232
%N A054557 Number of labeled pure 2-complexes on n nodes (0-simplexes) with 5 2-simplexes and 10 1-simplexes.
%C A054557 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 A054557 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 A054557 Vincenzo Librandi, <a href="/A054557/b054557.txt">Table of n, a(n) for n = 5..1000</a>
%H A054557 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F A054557 a(n) = 72*C(n, 5)+4392*C(n, 6) = n*(n-1)*(n-2)*(n-3)*(n-4)*(61*n-299)/10.
%F A054557 G.f.: 72*x^5*(1+60*x)/(1-x)^7. - _Colin Barker_, Jan 19 2012
%F A054557 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 A054557 CoefficientList[Series[72*(1+60*x)/(1-x)^7,{x,0,30}],x] (* _Vincenzo Librandi_, Apr 28 2012 *)
%o A054557 (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 A054557 nonn,easy
%O A054557 5,1
%A A054557 _Vladeta Jovovic_, Apr 10 2000
%E A054557 More terms from _James Sellers_, Apr 11 2000