A116508 a(n) = C( C(n,2), n).

1, 0, 0, 1, 15, 252, 5005, 116280, 3108105, 94143280, 3190187286, 119653565850, 4922879481520, 220495674290430, 10682005290753420, 556608279578340080, 31044058215401404845, 1845382436487682488000, 116475817125419611477660, 7779819801401934344268210

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

a(n) is the number of simple labeled graphs with n nodes and n edges. - Geoffrey Critzer, Nov 02 2014

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

Examples

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..370

Crossrefs

Main diagonal of A084546.

The unlabeled version is A001434, covering case A006649.

The connected case is A057500, unlabeled A001429.

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

The covering case is A367863.

A006125 counts graphs, A000088 unlabeled.

A006129 counts covering graphs, A002494 unlabeled.

A133686 counts graphs satisfying a strict AOC, connected A129271.

A367867 counts graphs contradicting a strict AOC, connected A140638.

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

Programs

Magma

[0] cat [(Binomial(Binomial(n+2, n), n+2)): n in [0..20]]; // Vincenzo Librandi, Nov 03 2014

Maple

a:= n-> binomial(binomial(n, 2), n):
seq(a(n), n=0..20);

Mathematica

nn = 18; f[x_, y_] :=
Sum[(1 + y)^Binomial[n, 2] x^n/n!, {n, 1, nn}]; Table[
n! Coefficient[Series[f[x, y], {x, 0, nn}], x^n y^n], {n, 1, nn}] (* Geoffrey Critzer, Nov 02 2014 *)
Table[Length[Subsets[Subsets[Range[n],{2}],{n}]],{n,0,5}] (* Gus Wiseman, Dec 22 2023 *)
Table[SeriesCoefficient[(1 + x)^(n*(n-1)/2), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 06 2025 *)

Python

from math import comb
def A116508(n): return comb(n*(n-1)>>1,n) # Chai Wah Wu, Jul 02 2024

Sage

[(binomial(binomial(n+2,n),n+2)) for n in range(-1, 17)] # Zerinvary Lajos, Nov 30 2009

Formula

a(n) ~ exp(n - 2) * n^(n - 1/2) / (sqrt(Pi) * 2^(n + 1/2)). - Vaclav Kotesovec, May 19 2020

a(n) = [x^n] (1+x)^(n*(n-1)/2). - Vaclav Kotesovec, Aug 06 2025

Extensions

a(0)=1 prepended by Alois P. Heinz, Feb 02 2024