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A000248 Number of forests with n nodes and height at most 1.
(Formerly M2857 N1148)
56
1, 1, 3, 10, 41, 196, 1057, 6322, 41393, 293608, 2237921, 18210094, 157329097, 1436630092, 13810863809, 139305550066, 1469959371233, 16184586405328, 185504221191745, 2208841954063318, 27272621155678841 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Equivalently, number of idempotent mappings f from a set of n elements into itself (i.e. satisfying f o f = f). - Robert FERREOL, Oct 11 2007

In other words, a(n) = number of idempotents in the full semigroup of maps from [1..n] to itself. [Tainiter]

a(n) is the number of ways to select a set partition of {1,2,...,n} and then designate one element in each block (cell) of the partition.

Let set B have cardinality n. Then a(n) is the number of functions f:D->C over all partitions {D,C} of B. See the example in the Example Section below. We note that f:empty set->B is designated as the null function, whereas f:B->empty set is undefined unless B itself is empty. [Dennis P. Walsh, Dec 05 2013]

REFERENCES

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 91.

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).

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.32(d).

LINKS

T. D. Noe, Table of n, a(n) for n=0..100

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 131

B. Harris and L. Schoenfeld, The number of idempotent elements in symmetric semigroups, J. Combin. Theory, 3 (1967), 122-135.

Bernard Harris and Lowell Schoenfeld, Asymptotic expansions for the coefficients of analytic functions, Illinois Journal of Mathematics, Volume 12, Issue 2 (1968), 264-277.

G. Helms, Pascalmatrix tetrated [From Gottfried Helms, Feb 04 2009]

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 117

Nate Kube and Frank Ruskey, Sequences That Satisfy a(n-a(n))=0, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.5.

J. Riordan, Forests of labeled trees, J. Combin. Theory, 5 (1968), 90-103.

M. Tainiter, A characterization of idempotents in semigroups, J. Combinat. Theory, 5 (1968), 370-373.

FORMULA

E.g.f.: exp(x*exp(x)).

G.f.: Sum_{k>=0} x^k/(1-k*x)^(k+1). - Vladeta Jovovic, Oct 25 2003

a(n) = Sum_{k=0..n} C(n,k)*(n-k)^k. [Paul D. Hanna, Jun 26 2009]

G.f.: G(0) where G(k) =  1 - x*(-1+2*k*x)^(2*k+1)/((x-1+2*k*x)^(2*k+2) - x*(x-1+2*k*x)^(4*k+4)/(x*(x-1+2*k*x)^(2*k+2) - (2*x-1+2*k*x)^(2*k+3)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 26 2013

E.g.f.: 1 + x/(1+x)*(G(0) - 1) where G(k) =  1 + exp(x)/(k+1)/(1-x/(x+(1)/G(k+1) )), (continued fraction). - Sergei N. Gladkovskii, Feb 04 2013

Recurrence: a(0)=1, a(n)=sum_{k=1..n} C(n-1,k-1)*k*a(n-k). - James East, Mar 30 2014

Asymptotics (Harris and Schoenfeld, 1968): a(n) ~ sqrt((r+1)/(2*Pi*(n+1)*(r^2+3*r+1))) * n! * exp((n+1)/(r+1)) / r^n, where r is the root of the equation r*(r+1)*exp(r) = n+1. - Vaclav Kotesovec, Jul 13 2014

EXAMPLE

a(3)=10 since, for B={1,2,3}, we have 10 functions: 1 function of the type f:empty set->B; 6 functions of the type f:{x}->B\{x}; and 3 functions of the type f:{x,y}->B\{x,y}. [Dennis P. Walsh, Dec 05 2013]

MAPLE

A000248 := proc(n) local k; add(k^(n-k)*binomial(n, k).k=0..n); end; # Robert FERREOL, Oct 11 2007

a:= proc(n) option remember; if n=0 then 1 else add (binomial (n-1, j) *(j+1) *a(n-1-j), j=0..n-1) fi end: seq (a(n), n=0..20); # Zerinvary Lajos, Mar 28 2009

MATHEMATICA

CoefficientList[Series[Exp[x Exp[x]], {x, 0, 20}], x]*Table[n!, {n, 0, 20}]

a[0] = 1; a[1] = 1; a[n_] := a[n] = a[n - 1] + Sum[(Binomial[n - 1, j] + (n - 1) Binomial[n - 2, j]) a[j], {j, 0, n - 2}]; Table[a[n], {n, 0, 20}] (* David Callan, Oct 04 2013 *)

Flatten[{1, Table[Sum[Binomial[n, k]*(n-k)^k, {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Jul 13 2014 *)

PROG

(PARI) a(n)=sum(k=0, n, binomial(n, k)*(n-k)^k); \\ Paul D. Hanna, Jun 26 2009

(PARI) x='x+O('x^66); Vec( serlaplace( exp(x*exp(x)) ) ) \\ Joerg Arndt, Oct 06 2013

CROSSREFS

First row of array A098697.

Row sums of A133399.

Column k=1 of A210725.

Column k=2 of A245501.

Sequence in context: A140046 A116540 A236407 * A245504 A030927 A002627

Adjacent sequences:  A000245 A000246 A000247 * A000249 A000250 A000251

KEYWORD

easy,nonn,nice

AUTHOR

N. J. A. Sloane, Simon Plouffe

STATUS

approved

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Last modified November 1 06:07 EDT 2014. Contains 248888 sequences.