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A134964 Number of different unlabeled n-node graphs with at most one cycle in each connected component. 6
1, 1, 2, 4, 9, 19, 46, 108, 273, 696, 1836, 4896, 13323, 36541, 101323, 282693, 793697, 2237982, 6335978, 17992622, 51235887, 146228734, 418181860, 1197972026, 3437159492, 9875198568, 28407202891, 81807809714, 235831978115 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n) is the number of pseudoforests on n nodes. - Eric W. Weisstein, Jun 11 2012

In first formula, for each partition P of n, we have a product over all parts p=1..n, that could arise in P. Since binomial(A005703(p) + a_p - 1, a_p) = 1, if a_p = 0, only the parts in P are considered. See example. - Washington Bomfim, Jun 27 2012

LINKS

Table of n, a(n) for n=0..28.

Eric Weisstein's World of Mathematics, Pseudoforest

Eric Weisstein's World of Mathematics, Frequency Representation

Wikipedia, Pseudoforest

FORMULA

a(0)=1. For n >= 1 a(n) = Sum of Product_{p=1..n, binomial(A005703(p) + a_p -1, a_p)} over all the partitions of n, having parts p with frequencies a_p.

Euler transform of A001429 + A000055. - Geoffrey Critzer, Oct 13 2012

EXAMPLE

a(5)=19 because from the formula, the partitions of 5, in frequency representation (1^a_1 2^a_2 ...), correspond respectively to

(1^5) => C(A005703(1) + 5 -1, 5) = 1,

(2^1 1^3) => C(A005703(2)+1-1, 1) * C(A005703(1)+3-1, 3) = C(1,1)*C(3,3)=1,

(2^2 1^1) => C(A005703(2)+2-1, 2) * C(A005703(1)+1-1, 1) = C(2,2)*C(1,1)=1,

(3^1 1^2) => C(A005703(3)+1-1, 1) * C(A005703(1)+2-1, 2) = C(2,1)*C(2,2)=2,

(3^1 2^1) => C(A005703(3)+1-1, 1) * C(A005703(2)+1-1, 1) = C(2,1)*C(1,1)=2,

(4^1 1^1) => C(A005703(4)+1-1, 1) * C(A005703(1)+1-1, 1) = C(4,1)*C(1,1)=4,

(5^1) => C(A005703(5)+1-1, 1) = C(8+1-1, 1) = 8;  and 1+1+1+2+2+4+8 = 19.

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

Cf. A005703 (number of pseudotrees), A137917 (number of maximal pseudoforests).

Sequence in context: A036613 A036614 A036718 * A318798 A318851 A052328

Adjacent sequences:  A134961 A134962 A134963 * A134965 A134966 A134967

KEYWORD

nonn

AUTHOR

Washington Bomfim, May 14 2008

EXTENSIONS

Edited by Washington Bomfim, Jun 27 2012

STATUS

approved

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Last modified August 13 22:53 EDT 2020. Contains 336473 sequences. (Running on oeis4.)