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 A133686 Number of labeled n-node graphs with at most one cycle in each connected component. 4
 1, 1, 2, 8, 57, 608, 8524, 145800, 2918123, 66617234, 1704913434, 48300128696, 1499864341015, 50648006463048, 1847622972848648, 72406232075624192, 3033607843748296089, 135313823447621913500, 6402077421524339766058, 320237988317922139148736 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..140 Wikipedia, Pseudoforest FORMULA 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). - Alois P. Heinz, Sep 15 2008 E.g.f.: sqrt(-LambertW(-x)/(x*(1+LambertW(-x))))*exp(-3/4 * LambertW(-x)^2). - Vladeta Jovovic, Sep 16 2008 E.g.f.: A(x)*B(x) where A(x) is the e.g.f. for A137916 and B(x) is the e.g.f. for A001858. - Geoffrey Critzer, Mar 23 2013 a(n) ~ 2^(-1/4) * Gamma(3/4) * exp(-1/4) * n^(n-1/4) / sqrt(Pi) * (1-7*Pi/(12*Gamma(3/4)^2*sqrt(n))). - Vaclav Kotesovec, Oct 08 2013 EXAMPLE 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 MAPLE cy:= proc(n) option remember; 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;       if k=0 then 1     elif k<0 or n add(T(n, k), k=0..n): seq(a(n), n=0..20); # Alois P. Heinz, Sep 15 2008 MATHEMATICA nn=20; t=Sum[n^(n-1)x^n/n!, {n, 1, nn}]; Range[0, nn]!CoefficientList[ Series[ Exp[t/2-3t^2/4]/(1-t)^(1/2), {x, 0, nn}], x] (* Geoffrey Critzer, Sep 05 2012 *) PROG (PARI) x='x+O('x^50); Vec(serlaplace(sqrt(-lambertw(-x)/(x*(1+ lambertw(-x))))*exp(-(3/4)*lambertw(-x)^2))) \\ G. C. Greubel, Nov 16 2017 CROSSREFS Cf. A129137, A005703, A000272, A057500, A129271, A134964, A137916, A001858. Row sums of triangle A144228. - Alois P. Heinz, Sep 15 2008 Cf. A137352. - Vladeta Jovovic, Sep 16 2008 Sequence in context: A153529 A153558 A027335 * A153525 A153553 A191481 Adjacent sequences:  A133683 A133684 A133685 * A133687 A133688 A133689 KEYWORD easy,nonn AUTHOR Washington Bomfim, May 12 2008 EXTENSIONS Corrected and extended by Alois P. Heinz and Vladeta Jovovic, Sep 15 2008 STATUS approved

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Last modified July 15 22:48 EDT 2019. Contains 325061 sequences. (Running on oeis4.)