A127136 Triangle read by rows: T(n,k) is the number of endofunctions on n objects with k components.

1, 2, 1, 4, 2, 1, 9, 7, 2, 1, 20, 17, 7, 2, 1, 51, 48, 21, 7, 2, 1, 125, 127, 60, 21, 7, 2, 1, 329, 352, 174, 65, 21, 7, 2, 1, 862, 963, 504, 190, 65, 21, 7, 2, 1, 2311, 2689, 1456, 570, 196, 65, 21, 7, 2, 1, 6217, 7496, 4212, 1684, 590, 196, 65, 21, 7, 2, 1

Offset: 1

Author

Franklin T. Adams-Watters, Jan 05 2007

Keywords

• nonn
• tabl

Comments

For k > n/2, T(n,k) = T(n-1,k-1). - Geoffrey Critzer, Oct 13 2012

Examples

			For n = 3, the 7 endofunctions are (1,2,3) -> (1,1,1), (1,1,2), (1,2,1), (2,1,1), (1,2,3), (1,3,2) and (2,3,1). The components are respectively 123, 123, 13|2, 123, 1|2|3, 1|23 and 123; the number of components is thus 1, 1, 2, 1, 2, 3, 2, 1, so row 3 is 4,2,1.
The triangle starts:
   1;
   2,  1;
   4,  2,  1;
   9,  7,  2,  1;
  20, 17,  7,  2,  1;
		

Crossrefs

Cf. A001372 (row sums), A127124, A127125, A002861 (first column).

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}];c=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,1,30}]],1];CoefficientList[Series[Product[1/(1-y x^i)^c[[i]],{i,1,nn-1}],{x,0,10}],{x,y}]//Grid  (* Geoffrey Critzer, Oct 13 2012, after code given by Robert A. Russell in A000081 *)

Formula

G.f.: Product_{k>=1} 1/(1 - x^k*y)^A002861(k).

Extensions

More terms from Geoffrey Critzer, Oct 13 2012

Corrected and extended by Alois P. Heinz, May 24 2013