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
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;
Programs
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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
Comments