A218334 - OEIS

%I #14 Jan 05 2021 02:40:16

%S 1,4,38,3,728,40,26704,730,15,1866256,20608,420,251548592,961324,

%T 12460,105,66296291072,79643424,484624,5040,34496488594816,

%U 12495365424,27712860,220500,945,35641657548953344,3844702446464,2619965040,11297440,69300,73354596206766622208,2341246104706784,458476648344,775542460,4192650,10395

%N Triangular array read by rows. T(n,k) is the number of simple labeled graphs on n nodes with no isolated nodes and exactly k components. n >= 2, 1 <= k < n/2.

%C Row sums are A006129.

%C For even n, T(n,n/2) = A001147(n) = (2n-1)!!.

%C Column k = 1 is A001187.

%H Alois P. Heinz, <a href="/A218334/b218334.txt">Rows n = 2..50, flattened</a>

%F E.g.f.: exp( y*log(A(x)) ) where A(x) is the e.g.f. for A006129.

%e                 1;

%e                 4;

%e                38,           3;

%e               728,          40;

%e             26704,         730,       15;

%e           1866256,       20608,      420;

%e         251548592,      961324,    12460,    105;

%e       66296291072,    79643424,   484624,   5040;

%e    34496488594816, 12495365424, 27712860, 220500, 945;

%t nn=12;a=Sum[2^Binomial[n,2]x^n/n!,{n,0,nn}];b=a/Exp[x];f[list_]:=Select[list,#>0&];Map[f,Drop[Range[0,nn]!CoefficientList[Series[Exp[y Log[b]],{x,0,nn}],{x,y}],2]]//Flatten

%K nonn,tabf

%O 2,2

%A _Geoffrey Critzer_, Oct 26 2012