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Search: id:A000248
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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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+0
25
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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
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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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 (ferreol(AT)mathcurve.com), Oct 11 2007
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.32(d).
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..100
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 117
G. Helms, Pascalmatrix tetrated [From Gottfried Helms (helms(AT)uni-kassel.de), Feb 04 2009]
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 131
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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 (vladeta(AT)eunet.rs), Oct 25 2003
a(n) = Sum_{k=0..n} C(n,k)*(n-k)^k. [From Paul D. Hanna (pauldhanna(AT)juno.com), Jun 26 2009]
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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 (ferreol(AT)mathcurve.com), Oct 11 2007
restart: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); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 28 2009]
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PROGRAM
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(PARI) a(n)=sum(k=0, n, binomial(n, k)*(n-k)^k) [From Paul D. Hanna (pauldhanna(AT)juno.com), Jun 26 2009]
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CROSSREFS
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First row of array A098697.
Row sums of A133399. [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Sep 19 2008]
Sequence in context: A151083 A140046 A116540 this_sequence A030927 A002627 A030802
Adjacent sequences: A000245 A000246 A000247 this_sequence A000249 A000250 A000251
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KEYWORD
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easy,nonn,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)
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