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A001858 Number of forests of trees on n labeled nodes.
(Formerly M1804 N0714)
24
1, 1, 2, 7, 38, 291, 2932, 36961, 561948, 10026505, 205608536, 4767440679, 123373203208, 3525630110107, 110284283006640, 3748357699560961, 137557910094840848, 5421179050350334929, 228359487335194570528, 10239206473040881277575, 486909744862576654283616 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
The number of integer lattice points in the permutation polytope of {1,2,...,n}. - Max Alekseyev, Jan 26 2010
Equals the number of score sequences for a tournament on n vertices. See Prop. 7 of the article by Bartels et al., or Example 3.1 in the article by Stanley. - David Radcliffe, Aug 02 2022
REFERENCES
B. Bollobas, Modern Graph Theory, Springer, 1998, p. 290.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..388 (first 101 terms from T. D. Noe)
J. E. Bartels, J. Mount, and D. J. A. Welsh, The polytope of win vectors. Annals of Combinatorics 1, 1-15 (1997). https://doi.org/10.1007/BF02558460
David Callan, A Combinatorial Derivation of the Number of Labeled Forests, J. Integer Seqs., Vol. 6, 2003.
Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions, thesis, 2002 [Local copy, with permission]
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 132.
Samuele Giraudo, Combalgebraic structures on decorated cliques, Formal Power Series and Algebraic Combinatorics, Séminaire Lotharingien de Combinatoire, 78B.15, 2017, p. 7, arXiv:1709.08416 [math.CO], 2017.
Anton Izosimov, Matrix polynomials, generalized Jacobians, and graphical zonotopes, arXiv:1506.05179 [math.AG], 2015.
Arun P. Mani and Rebecca J. Stones, Congruences for weighted number of labeled forests, INTEGERS 16 (2016). #A17.
J. Pitman, Coalescent Random Forests, J. Combin. Theory, A85 (1999), 165-193.
J. Riordan, Forests of labeled trees, J. Combin. Theory, 5 (1968), 90-103.
John Riordan and N. J. A. Sloane, Correspondence, 1974
R. Stanley, Decompositions of rational convex polytopes, Ann. Discrete Math 6.6 (1980): 333-342.
L. Takacs, On the number of distinct forests, SIAM J. Discrete Math., 3 (1990), 574-581.
E. M. Wright, A relationship between two sequences, Proc. London Math. Soc. (3) 17 (1967) 296-304.
FORMULA
E.g.f.: exp( Sum_{n>=1} n^(n-2)*x^n/n! ). This implies (by a theorem of Wright) that a(n) ~ exp(1/2)*n^(n-2). - N. J. A. Sloane, May 12 2008 [Corrected by Philippe Flajolet, Aug 17 2008]
E.g.f.: exp(T - T^2/2), where T = T(x) = Sum_{n>=1} n^(n-1)*x^n/n! is Euler's tree function (see A000169). - Len Smiley, Dec 12 2001
Shifts 1 place left under the hyperbinomial transform (cf. A088956). - Paul D. Hanna, Nov 03 2003
a(0) = 1, a(n) = Sum_{j=0..n-1} C(n-1,j) (j+1)^(j-1) a(n-1-j) if n>0. - Alois P. Heinz, Sep 15 2008
MAPLE
exp(x+x^2+add(n^(n-2)*x^n/n!, n=3..50));
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
binomial(n-1, j-1)*j^(j-2)*a(n-j), j=1..n))
end:
seq(a(n), n=0..20); # Alois P. Heinz, Sep 15 2008
# third Maple program:
F:= exp(-LambertW(-x)*(1+LambertW(-x)/2)):
S:= series(F, x, 51):
seq(coeff(S, x, j)*j!, j=0..50); # Robert Israel, May 21 2015
MATHEMATICA
nn=20; t=Sum[n^(n-1)x^n/n!, {n, 1, nn}]; Range[0, nn]!CoefficientList[ Series[Exp[t-t^2/2], {x, 0, nn}], x] (* Geoffrey Critzer, Sep 05 2012 *)
nmax = 20; CoefficientList[Series[-LambertW[-x]/(x*E^(LambertW[-x]^2/2)), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jul 19 2019 *)
PROG
(PARI) a(n)=if(n<0, 0, sum(m=0, n, sum(j=0, m, binomial(m, j)*binomial(n-1, n-m-j)*n^(n-m-j)*(m+j)!/(-2)^j)/m!)) /* Michael Somos, Aug 22 2002 */
CROSSREFS
Cf. A088956. Row sums of A138464. Cf. also A006572, A006573.
Sequence in context: A317985 A084552 A094664 * A233335 A000366 A341381
KEYWORD
nonn,easy,eigen
AUTHOR
EXTENSIONS
More terms from Michael Somos, Aug 22 2002
STATUS
approved

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