%I #35 Jul 10 2024 15:03:22
%S 1,2,7,42,386,4944,82160,1683218,40999516,1156626990,37060382822,
%T 1328700402564,52676695500313,2287415069586304,107943308165833912,
%U 5499354613856855290,300788453960472434648,17577197510240126035698,1092833166733915284972350
%N a(n) = Sum_{k=0..n} C(n*(n+1)/2,k).
%C Number of labeled loop-graphs with n vertices and at most n edges. - _Gus Wiseman_, Feb 14 2024
%H Harry J. Smith, <a href="/A066383/b066383.txt">Table of n, a(n) for n = 0..100</a>
%F a(n) = 2^(n*(n+1)/2) - binomial(n*(n+1)/2,n+1)*2F1(1,(-n^2+n+2)/2;n+2;-1) = A006125(n) - A116508(n+1) * 2F1(1,(-n^2+n+2)2;n+2;-1), where 2F1(a,b;c;x) is the hypergeometric function. - _Ilya Gutkovskiy_, May 06 2016
%F a(n) ~ exp(n) * n^(n - 1/2) / (sqrt(Pi) * 2^(n + 1/2)). - _Vaclav Kotesovec_, Feb 20 2024
%e From _Gus Wiseman_, Feb 14 2024: (Start)
%e The a(0) = 1 through a(2) = 7 loop-graphs (loops shown as singletons):
%e {} {} {}
%e {{1}} {{1}}
%e {{2}}
%e {{1,2}}
%e {{1},{2}}
%e {{1},{1,2}}
%e {{2},{1,2}}
%e (End)
%t f[n_] := Sum[Binomial[n (n + 1)/2, k], {k, 0, n}]; Array[f, 21, 0] (* _Vincenzo Librandi_, May 06 2016 *)
%t Table[Length[Select[Subsets[Subsets[Range[n],{1,2}]],Length[#]<=n&]],{n,0,5}] (* _Gus Wiseman_, Feb 14 2024 *)
%o (PARI) { for (n=0, 100, a=0; for (k=0, n, a+=binomial(n*(n + 1)/2, k)); write("b066383.txt", n, " ", a) ) } \\ _Harry J. Smith_, Feb 12 2010
%o (Python)
%o from math import comb
%o def A066383(n): return sum(comb(comb(n+1,2),k) for k in range(n+1)) # _Chai Wah Wu_, Jul 10 2024
%Y The case of equality is A014068, covering A368597.
%Y The loopless version is A369192, covering A369191.
%Y The covering case is A369194, minimal case A001862.
%Y Counting only covered vertices gives A369196, without loops A369193.
%Y The connected covering case is A369197, without loops A129271.
%Y The unlabeled version is A370168, covering A370169.
%Y A006125 counts simple graphs, unlabeled A000088.
%Y A006129 counts covering graphs, unlabeled A002494.
%Y A322661 counts covering loop-graphs, unlabeled A322700.
%Y Cf. A000085, A000666, A005703, A062740, A116508, A367862, A367863, A367916, A368927, A369141, A369199.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Dec 23 2001