A217955 Triangular array read by rows. T(n,k) is the number of unlabeled graphs on n nodes that have exactly k distinct components (n >= 1).

1, 1, 2, 1, 6, 2, 21, 8, 112, 28, 2, 853, 145, 7, 11117, 1022, 34, 261080, 12320, 181, 1, 11716571, 274785, 1266, 12, 1006700565, 12007355, 14106, 63, 164059830476, 1019030127, 293756, 407, 50335907869219, 165091859656, 12362198, 3023, 6, 29003487462848061, 50502058491413, 1032671168, 33035, 51, 31397381142761241960, 29054157815353374, 166176421788, 645086, 399, 63969560113225176176277, 31426486309136268658, 50672459139597, 25830118, 3113

Offset: 1

Author

Geoffrey Critzer, Oct 15 2012

Keywords

• nonn
• tabf

Comments

Row sums are A207828.

Column 1 is A001349.

Column 2 is A216785.

Column 3 is A058915.

Examples

			Triangle begins
         1;
         1;
         2,      1;
         6,      2;
        21,      8;
       112,     28,    2;
       853,    145,    7;
     11117,   1022,   34;
    261080,  12320,  181,  1;
  11716571, 274785, 1266, 12;
		

Programs

Mathematica

Needs["Combinatorica`"];max=20;A000088=Table[NumberOfGraphs[n],{n,0,max}];f[x_]=1-Product[1/(1-x^k)^a[k],{k,1,max}];a[0]=a[1]=a[2]=1;coes=CoefficientList[Series[f[x],{x,0,max}],x];sol=First[Solve[Thread[Rest[coes+A000088]== 0]]];cg=Table[a[n],{n,1,max}]/.sol;CoefficientList[Series[Product[(1+y x^i)^cg[[i]],{i,1,max}],{x,0,max}],{x,y}]//Grid (* after code by Jean-François Alcover in A001349 *)

Formula

O.g.f.: Product_{n>=1} (1 + y*x^n)^A001349(n).