OFFSET
0,3
COMMENTS
a(n) is the number of pseudoforests on n nodes. - Eric W. Weisstein, Jun 11 2012
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..500
F. Ruskey, Combinatorial Generation, Eq.(4.27), 2003
Eric Weisstein's World of Mathematics, Pseudoforest
Wikipedia, Pseudoforest
FORMULA
MATHEMATICA
Needs["Combinatorica`"];
nn=30; s[n_, k_]:=s[n, k]=a[n+1-k]+If[n<2k, 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}]; cu=Drop[Apply[Plus, Table[Take[CoefficientList[CycleIndex[DihedralGroup[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, 3, nn}]], 1]; t[n_, k_]:=t[n, k]=b[n+1-k]+If[n<2k, 0, t[n-k, k]]; b[1]=1; b[n_]:=b[n]=Sum[b[i]t[n-1, i]i, {i, 1, n-1}]/(n-1); ft=Table[b[i]-Sum[b[j]b[i-j], {j, 1, i/2}]+If[OddQ[i], 0, b[i/2](b[i/2]+1)/2], {i, 1, nn}];
CoefficientList[Series[Product[1/(1-x^i)^(cu[[i]]+ft[[i]]), {i, 1, nn-1}], {x, 0, nn}], x] (* Geoffrey Critzer, Oct 13 2012, after codes given by Robert A. Russell in A134964 and A000055 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Washington Bomfim, May 14 2008
EXTENSIONS
Edited by Washington Bomfim, Jun 27 2012
Terms a(29) and beyond from Andrew Howroyd, May 16 2021
STATUS
approved