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 A000949 Number of forests with n nodes and height at most 2. (Formerly M3021 N1223) 30
 1, 1, 3, 16, 101, 756, 6607, 65794, 733833, 9046648, 121961051, 1782690174, 28055070397, 472594822324, 8479144213191, 161340195463066, 3243707386310033, 68679247688467056, 1526976223741111987, 35557878951515668726, 865217354118762606021 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Equivalently, the number of mappings from a set of n elements into itself where f(f(x)) = f(f(f(x))). - Chad Brewbaker, Mar 26 2014 REFERENCES 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 T. D. Noe, Table of n, a(n) for n = 0..100 J. Riordan, Forests of labeled trees, J. Combin. Theory, 5 (1968), 90-103. FORMULA E.g.f.: exp(x*exp(x*exp(x))). a(n) = n!*sum(m=1..n-1, sum(k=1..n-m, (k^(n-m-k)*m^k)/(k!*(n-m-k)!))/m!)+1. - Vladimir Kruchinin, May 28 2011 EXAMPLE G.f. = 1 + x + 3*x^2 + 16*x^3 + 101*x^4 + 756*x^5 + 6607*x^6 + 65794*x^7 + ... - Michael Somos, Jul 03 2018 MATHEMATICA nn = 20; Range[0, nn]! CoefficientList[Series[Exp[x*Exp[x*Exp[x]]], {x, 0, nn}], x] (* T. D. Noe, Jun 21 2012 *) a[ n_] := If[ n < 0, 0, 1 + n! Sum[ Sum[ k^(n - m - k) m^k / (k! (n - m - k)!), {k, n - m}] / m!, {m, n - 1}]]; (* Michael Somos, Jul 03 2018 *) PROG (Maxima) a(n):=n!*sum(sum((k^(n-m-k)*m^k)/(k!*(n-m-k)!), k, 1, n-m)/m!, m, 1, n-1)+1; /* Vladimir Kruchinin, May 28 2011 */ (PARI) x='x+O('x^66); Vec(serlaplace(exp(x*exp(x*exp(x))))) /* show terms with a(0)=1 */ /* Joerg Arndt, May 28 2011 */ CROSSREFS Cf. A000248, A000950, A000951, A052512-A052514. Column k=2 of A210725. - Alois P. Heinz, Mar 15 2013 Sequence in context: A137572 A009151 A009007 * A091637 A278429 A207434 Adjacent sequences:  A000946 A000947 A000948 * A000950 A000951 A000952 KEYWORD nonn AUTHOR EXTENSIONS More terms from Vladeta Jovovic, Apr 07 2001 STATUS approved

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Last modified October 14 14:06 EDT 2019. Contains 328017 sequences. (Running on oeis4.)