Search: id:A000248
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%I A000248 M2857 N1148
%S A000248 1,1,3,10,41,196,1057,6322,41393,293608,2237921,18210094,
%T A000248 157329097,1436630092,13810863809,139305550066,1469959371233,
%U A000248 16184586405328,185504221191745,2208841954063318,27272621155678841
%N A000248 Number of forests with n nodes and height at most 1.
%C A000248 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
%D A000248 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000248 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000248 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 91.
%D A000248 B. Harris and L. Schoenfeld, The number of idempotent elements in symmetric 
               semigroups, J. Combin. Theory, 3 (1967), 122-135.
%D A000248 Nate Kube and Frank Ruskey, Sequences That Satisfy a(n-a(n))=0, Journal 
               of Integer Sequences, Vol. 8 (2005), Article 05.5.5.
%D A000248 J. Riordan, Forests of labeled trees, J. Combin. Theory, 5 (1968), 90-103.
%D A000248 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see 
               Problem 5.32(d).
%H A000248 T. D. Noe, <a href="b000248.txt">Table of n, a(n) for n=0..100</a>
%H A000248 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=117">
               Encyclopedia of Combinatorial Structures 117</a>
%H A000248 G. Helms, <a href="http://go.helms-net.de/math/tetdocs/PascalMatrixTetrated.pdf">
               Pascalmatrix tetrated</a> [From Gottfried Helms (helms(AT)uni-kassel.de), 
               Feb 04 2009]
%H A000248 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/
               Publications/books.html">Analytic Combinatorics</a>, 2009; see page 
               131
%F A000248 E.g.f.: exp(x*exp(x)).
%F A000248 G.f.: Sum_{k>=0} x^k/(1-k*x)^(k+1). - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Oct 25 2003
%F A000248 a(n) = Sum_{k=0..n} C(n,k)*(n-k)^k. [From Paul D. Hanna (pauldhanna(AT)juno.com), 
               Jun 26 2009]
%p A000248 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
%p A000248 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]
%o A000248 (PARI) a(n)=sum(k=0,n,binomial(n,k)*(n-k)^k) [From Paul D. Hanna (pauldhanna(AT)juno.com), 
               Jun 26 2009]
%Y A000248 First row of array A098697.
%Y A000248 Row sums of A133399. [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), 
               Sep 19 2008]
%Y A000248 Sequence in context: A151083 A140046 A116540 this_sequence A030927 A002627 
               A030802
%Y A000248 Adjacent sequences: A000245 A000246 A000247 this_sequence A000249 A000250 
               A000251
%K A000248 easy,nonn,nice
%O A000248 0,3
%A A000248 N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)

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