OFFSET
0,3
COMMENTS
Or, if the initial 0 is omitted, this is the binomial transform of the partition numbers p(1), p(2), ... = 1, 2, 3, 5, 7, 11, 15, 22, 30, ... (A000041 without the initial 1).
The most precise definition of this sequence is the Maple combstruct command given below. See the first Wieder link for further details.
Sequence appears to have a rational o.g.f. - Ralf Stephan, May 18 2007
For n>0, row sums of triangle A137151. - Gary W. Adamson, Jan 23 2008
LINKS
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.
Thomas Wieder, Expanded definitions of A103446 and A025168
FORMULA
O.g.f.: exp( Sum_{n>=1} sigma(n)*x^n/(1-x)^n/n ) - 1. - Paul D. Hanna, Apr 21 2010
O.g.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=1..n} binomial(n,k)*sigma(k) ) - 1. - Paul D. Hanna, Feb 04 2012
O.g.f. P(x/(1-x)), where P(x) is the o.g.f. for number of partitions (A000041) a(n)=sum_{k=1,n} ( binomial(n-1,k-1)*A000041(k)). - Vladimir Kruchinin, Aug 10 2010
a(n) ~ exp(Pi*sqrt(n/3) + Pi^2/24) * 2^(n-2) / (n*sqrt(3)). - Vaclav Kotesovec, Jun 25 2015
EXAMPLE
Let {} denote a set, [] a list and Z an unlabeled element.
a(3) = 8 because we have {[[Z]],[[Z]],[[Z]]}, {[[Z],[Z]],[[Z]]}, {[[Z],[Z],[Z]]}, {[[Z],[Z,Z]]}, {[[Z,Z],[Z]]}, {[[Z,Z]],[[Z]]}, {[[Z]],[[Z,Z]]}, {[[Z,Z,Z]]}.
MAPLE
with(combstruct); SubSetSeqU := [T, {T=Subst(U, S), S=Set(U, card>=1), U=Sequence(Z, card>=1)}, unlabeled]; [seq(count(SubSetSeqU, size=n), n=0..30)];
allstructs(SubSetSeq, size=3); # to get the structures for n=3 - this output is shown in the example lines.
MATHEMATICA
Flatten[{0, Table[Sum[Binomial[n-1, k]*PartitionsP[k+1], {k, 0, n-1}], {n, 1, 30}]}] (* Vaclav Kotesovec, Jun 25 2015 *)
PROG
(PARI) {a(n)=if(n<1, 0, polcoeff(exp(sum(m=1, n, sigma(m)*x^m/(1-x+x*O(x^n))^m/m)), n))} \\ Paul D. Hanna, Apr 21 2010
(PARI) {a(n)=if(n<1, 0, polcoeff(exp(sum(m=1, n, x^m/m*sum(k=1, m, binomial(m, k)*sigma(k)))+x*O(x^n)), n))} \\ Paul D. Hanna, Feb 04 2012
(PARI) Vec(1/eta('x/(1-'x)+O('x^66))) \\ Joerg Arndt, Jul 30 2011
CROSSREFS
KEYWORD
nonn
AUTHOR
Thomas Wieder, Feb 06 2005; revised Feb 20 2006
EXTENSIONS
I can confirm that the terms shown are the binomial transform of the partition sequence 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, ... (A000041 without the a(0) term). - N. J. A. Sloane, May 18 2007
STATUS
approved