0,2

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)];

Apart from the first term these numbers appear to show up in the middle of the even-numbered rows in triangle A154921. [From Mats Granvik (mats.granvik(AT)abo.fi), Feb 07 2009]

Thomas Wieder, Home Page.

Thomas Wieder, (Old) Home Page.

a(n) = binomial(2*n, n) * Sum_{k=1..n} k! * Stirling2(n, k) = binomial(2*n, n)*A000670(n).

Note: a_0(n) = Sum_{k=0..n} k! * Stirling2(n, k) * binomial(2*n, n) gives 1, 2, 18, 260, 5250, 136332, 4327092, 162309576, ...

a(0) = 1, a(n) = Sum_i=1^n Sum_k=1^i ((2*i)!/i!)*(1/k!)*((i-k)!/(2*(i-k))!) * a(i-k).

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].

main := proc(n::integer) local a, level, Summe; with(combinat): Summe := 0; for level from 1 to n do Summe := Summe + level! * stirling2(n, level) * binomial(2*n, n); ; end do; a := Summe; print("n, a(n):", n, a); end proc;

A099880 := proc(n::integer) local i, k, a, Summe; with(combinat): a[0] := 1; for i from 1 to n do Summe := 0; for k from 1 to i do Summe := Summe + ((2*i)!/i!)*(1/k!)*((i-k)!/(2*(i-k))!) * a[i-k]; end do; a[i] := Summe; end do; a[n] := Summe; print("A099880(n) = ", a[n]); end proc;

a:=n->sum(binomial(2*n, n)*(stirling2(n, k))*k!, k=1..n):seq(a(n), n=0..16); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 19 2006

f[n_] := Sum[l! StirlingS2[n, l] Binomial[2n, n], {l, n}]; Table[ f[n], {n, 0, 16}] (from Robert G. Wilson v Nov 04 2004)

Cf. A000670, A075729, A003480.

Sequence in context: A109517 A143138 A151362 this_sequence A141009 A143154 A032037

Adjacent sequences: A099877 A099878 A099879 this_sequence A099881 A099882 A099883

nonn

Thomas Wieder (wieder.thomas(AT)t-online.de), Nov 02 2004

More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Nov 04 2004