A066383 a(n) = Sum_{k=0..n} C(n*(n+1)/2,k).

1, 2, 7, 42, 386, 4944, 82160, 1683218, 40999516, 1156626990, 37060382822, 1328700402564, 52676695500313, 2287415069586304, 107943308165833912, 5499354613856855290, 300788453960472434648, 17577197510240126035698, 1092833166733915284972350

Offset

0,2

Comments

Number of labeled loop-graphs with n vertices and at most n edges. - Gus Wiseman, Feb 14 2024

Links

Harry J. Smith, Table of n, a(n) for n = 0..100

Formula

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

Example

From Gus Wiseman, Feb 14 2024: (Start)
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)

Mathematica

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 *)

Prog

(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
(Python)
from math import comb
def A066383(n): return sum(comb(comb(n+1, 2), k) for k in range(n+1)) # Chai Wah Wu, Jul 10 2024

Crossrefs

The case of equality is A014068, covering A368597.

The loopless version is A369192, covering A369191.

The covering case is A369194, minimal case A001862.

Counting only covered vertices gives A369196, without loops A369193.

The connected covering case is A369197, without loops A129271.

The unlabeled version is A370168, covering A370169.

A006125 counts simple graphs, unlabeled A000088.

A006129 counts covering graphs, unlabeled A002494.

A322661 counts covering loop-graphs, unlabeled A322700.

Cf. A000085, A000666, A005703, A062740, A116508, A367862, A367863, A367916, A368927, A369141, A369199.

Sequence in context: A359717 A038052 A371079 * A011802 A007065 A352258

Adjacent sequences: A066380 A066381 A066382 * A066384 A066385 A066386

Keyword

nonn

Author

N. J. A. Sloane, Dec 23 2001

Status

approved