login
Total number of n-colorings of all graphs on n labeled nodes.
3

%I #15 Jan 22 2025 17:01:37

%S 1,1,6,123,7108,1058885,386056326,332908216711,662759754883080,

%T 2991536714452469769,30189087071961437767690,

%U 673536638307789642838763531,32919693104791033024939149066252,3498056629389633452501822564131061773,802931613320922331646386276441560583143438

%N Total number of n-colorings of all graphs on n labeled nodes.

%H Alois P. Heinz, <a href="/A372920/b372920.txt">Table of n, a(n) for n = 0..76</a>

%F a(n) = A322280(n,n).

%F a(n) == n (mod 2) for n >= 1.

%t nmax = 76;

%t a[n_] := 2^(n(n-1)/2)*n!*SeriesCoefficient[Sum[x^i/(2^(i(i-1)/2)*i!), {i, 0, nmax}]^n, {x, 0, n}];

%t Table[a[n],{n,0,nmax}] (* _Jean-François Alcover_, May 28 2024 *)

%Y Main diagonal of A322280.

%Y Cf. A000035, A006125.

%K nonn

%O 0,3

%A _Alois P. Heinz_, May 16 2024