A054592 Number of disconnected labeled graphs with n nodes.

0, 0, 1, 4, 26, 296, 6064, 230896, 16886864, 2423185664, 687883494016, 387139470010624, 432380088071584256, 959252253993204724736, 4231267540316814507357184, 37138269572860613284747227136, 649037449132671895468073434916864, 22596879313063804832510513481261154304

Offset: 0

Author

Vladeta Jovovic, Apr 15 2000

Keywords

• easy
• nonn

Crossrefs

Cf. A001187, A000719, A360604.

Programs

Maple

upto := 18: g := add(2^binomial(n, 2) * x^n / n!, n = 0..upto+1):
ser := series(g - log(g) - 1, x, upto+1):
seq(n!*coeff(ser, x, n), n = 0..upto); # Peter Luschny, Feb 24 2023

Mathematica

g=Sum[2^Binomial[n,2]x^n/n!,{n,0,20}]; Range[0,20]! CoefficientList[Series[g-Log[g]-1,{x,0,20}],x]  (* Geoffrey Critzer, Nov 11 2011 *)

Formula

a(n) = 2^binomial(n, 2) - A001187(n).

a(n) = n!*[x^n](g - log(g) - 1) where g = Sum_{n>=0} 2^binomial(n, 2) * x^n / n!. - Geoffrey Critzer, Nov 11 2011

a(n) = Sum_{k=0..n-1} A360604(n, k) * A001187(k). - Peter Luschny, Feb 24 2023

Extensions

Edited and extended with a(0) = 0 by Peter Luschny, Feb 24 2023