A217897 Triangular array read by rows. T(n,k) is the number of unlabeled functions on n nodes that have exactly k fixed points, n >= 0, 0 <= k <= n.

1, 0, 1, 1, 1, 1, 2, 3, 1, 1, 6, 7, 4, 1, 1, 13, 19, 9, 4, 1, 1, 40, 47, 27, 10, 4, 1, 1, 100, 130, 68, 29, 10, 4, 1, 1, 291, 343, 195, 76, 30, 10, 4, 1, 1, 797, 951, 523, 220, 78, 30, 10, 4, 1, 1, 2273, 2615, 1477, 600, 228, 79, 30, 10, 4, 1, 1, 6389, 7318, 4096, 1708, 625, 230, 79, 30, 10, 4, 1, 1

Offset: 0

Author

Geoffrey Critzer, Oct 14 2012

Keywords

• nonn
• tabl

Comments

Row sums are A001372;

Column for k=0 is A001373;

Column for k=1 is A001372. (offset)

Examples

			Triangle begins:
     1;
     0,    1;
     1,    1,    1;
     2,    3,    1,   1;
     6,    7,    4,   1,   1;
    13,   19,    9,   4,   1,  1;
    40,   47,   27,  10,   4,  1,  1;
   100,  130,   68,  29,  10,  4,  1,  1;
   291,  343,  195,  76,  30, 10,  4,  1, 1;
   797,  951,  523, 220,  78, 30, 10,  4, 1, 1;
  2273, 2615, 1477, 600, 228, 79, 30, 10, 4, 1, 1;
		

Links

• N. J. A. Sloane, Illustration of initial terms of A001372

Programs

Mathematica

Needs["Combinatorica`"]; nn=30;s[n_,k_]:=s[n,k]=a[n+1-k]+If[n<2 k,0,s[n-k,k]];a[1]=1;a[n_]:=a[n]=Sum[a[i] s[n-1,i] i,{i,1,n-1}]/(n-1);rt=Table[a[i],{i,1,nn}];cfd=Drop[Apply[Plus,Table[Take[CoefficientList[CycleIndex[CyclicGroup[n],s]/.Table[s[j]->Table[Sum[rt[[i]] x^(k*i),{i,1,nn}],{k,1,nn}][[j]],{j,1,nn}],x],nn],{n,2,30}]],1]; CoefficientList[Series[Product[1/(1-x^i)^cfd[[i]]/(1-y x^i)^rt[[i]],{i,1,nn-1}],{x,0,10}],{x,y}]//Grid (* after code given by Robert A. Russell in A000081 *)

Formula

O.g.f.: Product_{n>=1} 1/((1-x^n)^A002862(n) * (1 - y*x^n)^A000081(n) ).