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A000248
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Number of forests with n nodes and height at most 1.
(Formerly M2857 N1148)
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38
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1, 1, 3, 10, 41, 196, 1057, 6322, 41393, 293608, 2237921, 18210094, 157329097, 1436630092, 13810863809, 139305550066, 1469959371233, 16184586405328, 185504221191745, 2208841954063318, 27272621155678841
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OFFSET
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0,3
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COMMENTS
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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.
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 91.
B. Harris and L. Schoenfeld, The number of idempotent elements in symmetric semigroups, J. Combin. Theory, 3 (1967), 122-135.
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.
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).
M. Tainiter, A characterization of idempotents in semigroups, J. Combinat. Theory, 5 (1968), 370-373. - From N. J. A. Sloane, May 06 2012
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..100
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 131
G. Helms, Pascalmatrix tetrated [From Gottfried Helms, Feb 04 2009]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 117
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FORMULA
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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. [From 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) )); (recursively defined 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) )), recursively defined continued fraction. - Sergei N. Gladkovskii, Feb 04 2013
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MAPLE
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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 (zerinvarylajos(AT)yahoo.com), Mar 28 2009
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MATHEMATICA
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CoefficientList[Series[Exp[x Exp[x]], {x, 0, 20}], x]*Table[n!, {n, 0, 20}]
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PROG
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(PARI) a(n)=sum(k=0, n, binomial(n, k)*(n-k)^k) [From Paul D. Hanna, Jun 26 2009]
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CROSSREFS
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First row of array A098697.
Row sums of A133399. - Alois P. Heinz, Sep 19 2008
Column k=1 of A210725. - Alois P. Heinz, Mar 15 2013
Sequence in context: A151083 A140046 A116540 * A030927 A002627 A030802
Adjacent sequences: A000245 A000246 A000247 * A000249 A000250 A000251
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KEYWORD
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easy,nonn,nice,changed
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AUTHOR
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N. J. A. Sloane, Simon Plouffe
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STATUS
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approved
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