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

%I A127136 #28 Jan 26 2025 17:24:30
%S A127136 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,
%T A127136 329,352,174,65,21,7,2,1,862,963,504,190,65,21,7,2,1,2311,2689,1456,
%U A127136 570,196,65,21,7,2,1,6217,7496,4212,1684,590,196,65,21,7,2,1
%N A127136 Triangle read by rows: T(n,k) is the number of endofunctions on n objects with k components.
%C A127136 For k > n/2, T(n,k) = T(n-1,k-1). - _Geoffrey Critzer_, Oct 13 2012
%F A127136 G.f.: Product_{k>=1} 1/(1 - x^k*y)^A002861(k).
%e A127136 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.
%e A127136 The triangle starts:
%e A127136    1;
%e A127136    2,  1;
%e A127136    4,  2,  1;
%e A127136    9,  7,  2,  1;
%e A127136   20, 17,  7,  2,  1;
%t A127136 Needs["Combinatorica`"];
%t A127136 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 *)
%Y A127136 Cf. A001372 (row sums), A127124, A127125, A002861 (first column).
%K A127136 nonn,tabl
%O A127136 1,2
%A A127136 _Franklin T. Adams-Watters_, Jan 05 2007
%E A127136 More terms from _Geoffrey Critzer_, Oct 13 2012
%E A127136 Corrected and extended by _Alois P. Heinz_, May 24 2013