%I #32 Jul 02 2024 14:49:09

%S 1,0,0,1,15,252,5005,116280,3108105,94143280,3190187286,119653565850,

%T 4922879481520,220495674290430,10682005290753420,556608279578340080,

%U 31044058215401404845,1845382436487682488000,116475817125419611477660,7779819801401934344268210

%N a(n) = C( C(n,2), n).

%C a(n) is the number of simple labeled graphs with n nodes and n edges. - _Geoffrey Critzer_, Nov 02 2014

%C These graphs are not necessarily covering, but the covering case is A367863, unlabeled A006649, and the unlabeled version is A001434. - _Gus Wiseman_, Dec 22 2023

%H Alois P. Heinz, <a href="/A116508/b116508.txt">Table of n, a(n) for n = 0..370</a>

%F a(n) ~ exp(n - 2) * n^(n - 1/2) / (sqrt(Pi) * 2^(n + 1/2)). - _Vaclav Kotesovec_, May 19 2020

%e a(5) = C(C(5,2),5) = C(10,5) = 252.

%p a:= n-> binomial(binomial(n, 2), n):

%p seq(a(n), n=0..20);

%t nn = 18; f[x_, y_] :=

%t Sum[(1 + y)^Binomial[n, 2] x^n/n!, {n, 1, nn}]; Table[

%t n! Coefficient[Series[f[x, y], {x, 0, nn}], x^n y^n], {n, 1, nn}] (* _Geoffrey Critzer_, Nov 02 2014 *)

%t Table[Length[Subsets[Subsets[Range[n],{2}],{n}]],{n,0,5}] (* _Gus Wiseman_, Dec 22 2023 *)

%o (Sage) [(binomial(binomial(n+2,n),n+2)) for n in range(-1, 17)] # _Zerinvary Lajos_, Nov 30 2009

%o (Magma) [0] cat [(Binomial(Binomial(n+2, n), n+2)): n in [0..20]]; // _Vincenzo Librandi_, Nov 03 2014

%o (Python)

%o from math import comb

%o def A116508(n): return comb(n*(n-1)>>1,n) # _Chai Wah Wu_, Jul 02 2024

%Y Cf. A084546.

%Y The unlabeled version is A001434, covering case A006649.

%Y The connected case is A057500, unlabeled A001429.

%Y For set-systems we have A136556, covering case A054780.

%Y The covering case is A367863.

%Y A006125 counts graphs, A000088 unlabeled.

%Y A006129 counts covering graphs, A002494 unlabeled.

%Y A133686 counts graphs satisfying a strict AOC, connected A129271.

%Y A367867 counts graphs contradicting a strict AOC, connected A140638.

%Y Cf. A001187, A003465, A143543, A305000, A367916, A367917.

%K easy,nonn

%O 0,5

%A Christopher Hanusa (chanusa(AT)math.binghamton.edu), Mar 21 2006

%E a(0)=1 prepended by _Alois P. Heinz_, Feb 02 2024