A000949 Number of forests with n nodes and height at most 2. (Formerly M3021 N1223)

1, 1, 3, 16, 101, 756, 6607, 65794, 733833, 9046648, 121961051, 1782690174, 28055070397, 472594822324, 8479144213191, 161340195463066, 3243707386310033, 68679247688467056, 1526976223741111987, 35557878951515668726, 865217354118762606021

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: A009151 A009007 A394123 * A390014 A390215 A091637

Adjacent sequences: A000946 A000947 A000948 * A000950 A000951 A000952

Keyword

nonn,changed

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic, Apr 07 2001

Status

approved