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A066383
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a(n) = Sum_{k=0..n} C(n*(n+1)/2,k).
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17
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1, 2, 7, 42, 386, 4944, 82160, 1683218, 40999516, 1156626990, 37060382822, 1328700402564, 52676695500313, 2287415069586304, 107943308165833912, 5499354613856855290, 300788453960472434648, 17577197510240126035698, 1092833166733915284972350
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OFFSET
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0,2
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COMMENTS
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Number of labeled loop-graphs with n vertices and at most n edges. - Gus Wiseman, Feb 14 2024
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LINKS
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FORMULA
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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
a(n) ~ exp(n) * n^(n - 1/2) / (sqrt(Pi) * 2^(n + 1/2)). - Vaclav Kotesovec, Feb 20 2024
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EXAMPLE
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The a(0) = 1 through a(2) = 7 loop-graphs (loops shown as singletons):
{} {} {}
{{1}} {{1}}
{{2}}
{{1,2}}
{{1},{2}}
{{1},{1,2}}
{{2},{1,2}}
(End)
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MATHEMATICA
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f[n_] := Sum[Binomial[n (n + 1)/2, k], {k, 0, n}]; Array[f, 21, 0] (* Vincenzo Librandi, May 06 2016 *)
Table[Length[Select[Subsets[Subsets[Range[n], {1, 2}]], Length[#]<=n&]], {n, 0, 5}] (* Gus Wiseman, Feb 14 2024 *)
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PROG
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(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
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CROSSREFS
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Cf. A000085, A000666, A005703, A062740, A116508, A367862, A367863, A367916, A368927, A369141, A369199.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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