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A217859 Triangular array read by rows. T(n,k) is the number of functions on n unlabeled nodes that have exactly k unique components (n >= 1, k >= 1).

%I A217859 #25 Oct 02 2017 02:17:20
%S A217859 1,3,5,2,12,7,21,25,1,58,63,9,126,178,39,341,466,140,4,867,1253,470,
%T A217859 25,2334,3418,1431,135,6218,9365,4358,544,6,17016,25924,12871,2042,50,
%U A217859 46351,72207,37993,7056,291,127842,202345,111142,23483,1383,4,353297,568822,325359,75701,5754,60
%N A217859 Triangular array read by rows. T(n,k) is the number of functions on n unlabeled nodes that have exactly k unique components (n >= 1, k >= 1).
%C A217859 Row sums are A001372.
%C A217859 T(n,1) = A002861(n) + 1 when n is prime (counts connected functions and the identity function).
%H A217859 N. J. A. Sloane, <a href="https://oeis.org/a001372.gif">Illustration of initial terms</a>
%F A217859 O.g.f.: Product_{n>=1} ((y*x^n - x^n + 1)/(1 - x^n))^A002861(n).
%e A217859 Triangle begins:
%e A217859        1;
%e A217859        3,
%e A217859        5,      2;
%e A217859       12,      7;
%e A217859       21,     25,      1;
%e A217859       58,     63,      9;
%e A217859      126,    178,     39;
%e A217859      341,    466,    140,     4;
%e A217859      867,   1253,    470,    25;
%e A217859     2334,   3418,   1431,   135;
%e A217859     6218,   9365,   4358,   544,    6;
%e A217859    17016,  25924,  12871,  2042,   50;
%e A217859    46351,  72207,  37993,  7056,  291;
%e A217859   127842, 202345, 111142, 23483, 1383,  4;
%e A217859   353297, 568822, 325359, 75701, 5754, 60;
%e A217859 T(3,2)=2 because (in the link) the third and the fifth digraphs on 3 nodes are composed of 2 unique components.
%t A217859 Needs["Combinatorica`"];
%t A217859 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[((y x^i +1-x^i)/(1-x^i))^c[[i]],{i,1,nn-1}],{x,0,15}],{x,y}]//Grid
%t A217859 (* after code given by _Robert A. Russell_ in A000081 *)
%K A217859 nonn,tabf
%O A217859 1,2
%A A217859 _Geoffrey Critzer_, Oct 13 2012