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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.

%I A217897 #19 Oct 02 2017 02:17:23
%S A217897 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,
%T A217897 130,68,29,10,4,1,1,291,343,195,76,30,10,4,1,1,797,951,523,220,78,30,
%U A217897 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
%N 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.
%C A217897 Row sums are A001372;
%C A217897   Column for k=0 is A001373;
%C A217897   Column for k=1 is A001372. (offset)
%H A217897 N. J. A. Sloane, <a href="/A001372/a001372.gif">Illustration of initial terms of A001372</a>
%F A217897 O.g.f.: Product_{n>=1} 1/((1-x^n)^A002862(n) * (1 - y*x^n)^A000081(n) ).
%e A217897 Triangle begins:
%e A217897      1;
%e A217897      0,    1;
%e A217897      1,    1,    1;
%e A217897      2,    3,    1,   1;
%e A217897      6,    7,    4,   1,   1;
%e A217897     13,   19,    9,   4,   1,  1;
%e A217897     40,   47,   27,  10,   4,  1,  1;
%e A217897    100,  130,   68,  29,  10,  4,  1,  1;
%e A217897    291,  343,  195,  76,  30, 10,  4,  1, 1;
%e A217897    797,  951,  523, 220,  78, 30, 10,  4, 1, 1;
%e A217897   2273, 2615, 1477, 600, 228, 79, 30, 10, 4, 1, 1;
%t A217897 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 *)
%K A217897 nonn,tabl
%O A217897 0,7
%A A217897 _Geoffrey Critzer_, Oct 14 2012