OFFSET
0,2
COMMENTS
The unlabeled case seems to be given by A003480, which can be generated by the following combstruct command: SeqUnionU := [S, {S=Sequence(Set(U,card>=1), card>=1), U=Union(a,b), a=Atom, b=Atom},unlabeled]; [seq(count(SeqUnionU, size=n), n=0..20)]; .
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..348
FORMULA
a(n) = binomial(2*n, n) * Sum_{k=0..n} k! * Stirling2(n, k).
a(n) = binomial(2*n, n) * A000670(n).
a(n) = A154921(2n,n). - Mats Granvik, Feb 07 2009
EXAMPLE
Let a[1], a[2],...,a[n] and b[1],b[2],...,b[n] denote two kinds "a" and "b" of labeled elements where each kind as n elements in total.
Let ":" denote a level, e.g., if the elements a[1] and a[2] are on level L=1 and the element b[1] is on level L=2 then a[1]a[2]:b[1] is a preferrential arrangement (a simple hierarchy) with two levels.
Then for n=2 we have a(2) = 18 arrangements: a[1]a[2]; a[1]:a[2]; a[2]:a[1]; a[1]b[1]; a[1]:b[1]; b[1]:a[1]; a[1]b[2]; a[1]:b[2]; b[2]:a[1]; a[2]b[1]; a[2]:b[1]; b[1]:a[2]; a[2]b[2]; a[2]:b[2]; b[2]:a[2]; b[1]b[2]; b[1]:b[2]; b[2]:b[1].
MAPLE
a:=n-> add(binomial(2*n, n)*(Stirling2(n, k))*k!, k=0..n): seq(a(n), n=0..16); # Zerinvary Lajos, Oct 19 2006
# second Maple program:
b:= proc(n) b(n):= `if`(n=0, 1, add(b(n-j)/j!, j=1..n)) end:
a:= n-> b(n)*(2*n)!/n!:
seq(a(n), n=0..20); # Alois P. Heinz, Feb 03 2019
MATHEMATICA
f[n_] := Sum[l! StirlingS2[n, l] Binomial[2n, n], {l, n}]; Table[ f[n], {n, 0, 16}] (* Robert G. Wilson v, Nov 04 2004 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Thomas Wieder, Nov 02 2004
EXTENSIONS
More terms from Robert G. Wilson v, Nov 04 2004
a(0) corrected and edited by Alois P. Heinz, Feb 03 2019
STATUS
approved