0,3

The total number of those graphs of order 5 is 608. The number of forests of trees on n labeled nodes of order 5 is 291, so the majority of the graphs of that kind have one or more unicycles.

Wikipedia, Pseudoforest

a(0) = 1; for n >=1, a(n) = Sum of n!prod_{j=1}^n\{ frac{ A129271(j)^{c_j} } { j!^{c_j}c_j! } } over all the partitions of n, c_1 + 2c_2 + ... + nc_n; c_1, c_2, ..., c_n >= 0.

a(n) = Sum_{k=0..n} A144228(n,k). [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Sep 15 2008]

E.g.f.: sqrt(-LambertW(-x)/(x*(1+LambertW(-x))))*exp(-3/4*LambertW(-x)^2). [From Vladeta Jovovic (vladeta(AT)eunet.yu), Sep 16 2008]

Below we see the 7 partitions of n=5 in the form c_1 + 2c_2 + ... + nc_n

followed by the corresponding number of graphs. We consider the values of A129271(j) given by the table

j|1|2|3| 4| 5|

----+-+-+-+--+---+

a(j)|1|1|4|31|347|

1*5 -> 5!1^5 / (1!^5 * 5!) = 1

2*1 + 1*3 -> 5!1^1 * 1^3 / (2!^1 * 1! * 1!^3 * 3!) = 10

2*2 + 1*1 -> 5!1^2 * 1^1 / (2!^2 * 2! * 1!^1 * 1!) = 15

3*1 + 1*2 -> 5!4^1 * 1^2 / (3!^1 * 1! * 1!^2 * 2!) = 40

3*1 + 2*1 -> 5!4^1 * 1^1 / (3!^1 * 1! * 2!^1 * 1!) = 40

4*1 + 1*1 -> 5!31^1 * 1^1 / (4!^1 * 1! * 1!^1 * 1!) = 155

5*1 -> 5!347^1 / (5!^1 * 1!) = 347

Total 608

cy:= proc(n) option remember; local t; binomial(n-1, 2) *add ((n-3)! /(n-2-t)! *n^(n-2-t), t=1..n-2) end: T:= proc(n, k) option remember; local j; if k=0 then 1 elif k<0 or n<k then 0 else add (binomial (n-1, j) *((j+1)^(j-1) *T(n-j-1, k-j) +cy(j+1) *T(n-j-1, k-j-1)), j=0..k) fi end: a:= n-> add (T(n, k), k=0..n): seq (a(n), n=0..20); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Sep 15 2008]

Cf. A129137, A005703, A000272, A057500, A129271, A134964, A137916, A001858.

Row sums of triangle A144228. [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Sep 15 2008]

Cf. A137352, A144228. [From Vladeta Jovovic (vladeta(AT)eunet.yu), Sep 16 2008]

Sequence in context: A084872 A113725 A027335 this_sequence A007347 A063074 A005804

Adjacent sequences: A133683 A133684 A133685 this_sequence A133687 A133688 A133689

easy,nonn

Washington Bomfim (webonfim(AT)bol.com.br), May 12 2008

Corrected and extended by Alois P. Heinz (heinz(AT)hs-heilbronn.de) and Vladeta Jovovic (vladeta(AT)eunet.yu), Sep 15 2008