0,3

Lists of lists of sets.

T. S. Motzkin, Sorting numbers ...: for a link to this paper see A000262.

T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176.

N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.

Index entries for sequences related to rooted trees

E.g.f.: (2-e^x)/(3-2*e^x).

a(n) is asymptotic to (1/6)*n!/log(3/2)^(n+1). - Benoit Cloitre, Jan 30 2003

For m-level trees (m>1), e.g.f. is (m-1-(m-2)*e^x)/(m-(m-1)*e^x), and number of trees is 1/(m*(m-1))*sum(k>=0, (1-1/m)^k*k^n). Here m=3, so a(n)=(1/6)*sum(k>=0, (2/3)^k*k^n) (for n>0). - Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 30 2003

a(n) = Sum_{k=1..n} Stirling2(n, k)*k!*2^(k-1). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Sep 28 2003

Recurrence : a(n+1) = 1 + 2*sum { j=1, n, (binomial(n+1, j)*a(j) } - Jon Perry (perry(AT)globalnet.co.uk), Apr 25 2005

With p(n) = the number of integer partitions of n, p(i) = the number of parts of the i-th partition of n, d(i) = the number of different parts of the i-th partition of n, p(j, i) = the j-th part of the i-th partition of n, m(i, j) = multiplicity of the j-th part of the i-th partition of n, sum_{i=1}^{p(n)} = sum over i, and prod_{j=1}^{d(i)} = product over j one has: a(n)=sum_{i=1}^{p(n)}(n!/(prod_{j=1}^{p(i)}p(i, j)!))*(p(i)!/(prod_{j=1}^{d(i)} m(i, j)!))*2^(p(i)-1) - Thomas Wieder (wieder.thomas(AT)t-online.de), May 18 2005

with(combstruct); SeqSeqSetL := [T, {T=Sequence(S), S=Sequence(U, card >= 1), U=Set(Z, card >=1)}, labeled];

(PARI) a(n)=n!*if(n<0, 0, polcoeff((2-exp(x))/(3-2*exp(x))+O(x^(n+1)), n))

Cf. A000670, A050352-A050359.

Equals 1/2 * A004123(n) for n>0.

Sequence in context: A112698 A025168 A084358 this_sequence A129137 A055869 A112937

Adjacent sequences: A050348 A050349 A050350 this_sequence A050352 A050353 A050354

nonn

Christian G. Bower (bowerc(AT)usa.net), Oct 15 1999.