A217896 - OEIS

%I #17 Oct 11 2017 05:09:22

%S 0,1,2,5,13,34,90,243,660,1818,5045,14102,39639,111982,317533,903464,

%T 2577724,7372542,21130127,60672017,174492633,502568607,1449360241,

%U 4184719174,12095325486,34993693260,101332159421,293669741860,851722291650,2471948910379,6824540110584

%N Number of unlabeled functions on n nodes that have at least one fixed point.

%C See A001372 and A001373

%t 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];cf=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];fd=CoefficientList[Series[Product[1/(1-x^i)^cfd[[i]],{i,1,nn-1}],{x,0,nn}],x];f=CoefficientList[Series[Product[1/(1-x^i)^cf[[i]],{i,1,nn-1}],{x,0,nn}],x];f-fd (* _Geoffrey Critzer_, Oct 14 2012, after code given by _Robert A. Russell_ in A000081 *)

%Y Cf. A001372 and A001373.

%K nonn

%O 0,3

%A _Geoffrey Critzer_, Oct 14 2012