1,2

A035098 and A000070 are near the two ends of a spectrum. Another way to look at A000070 is as the number of partitions of an n-multiset with multiplicities n-1, 1.

The very ends are the number of partitions and the Stirling numbers of the second kind, which count the n-multiset partitions with multiplicities n and 1,1,1,...,1, respectively.

Intermediate sequences are the number of ways of partitioning an n-multiset with multiplicities some partition of n.

Sum_{k=0..n} Stirling2(n, k)*((k+1)*(k+2)/2+1). E.g.f.: 1/2*(1+exp(x))^2*exp(exp(x)-1). (1/2)*(Bell(n)+Bell(n+1)+Bell(n+2)). - Vladeta Jovovic (vladeta(AT)eunet.rs), Sep 23 2003

a(3)=4 because there are 4 ways to partition the multiset {1,2,2} (with multiplicities {1,2}): {{1,2,2}} {{1,2},{2}} {{1},{2,2}} {{1},{2},{2}}.

with (combinat):a:=n->floor(1/2*(bell(n)+bell(n+1)+bell(n+2))): seq(a(n), n=-1..21); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 07 2007

The author has a Mathematica program to calculate these.

Cf. A000070.

Cf. A000110, A059606.

Sequence in context: A071794 A107378 A086611 this_sequence A138301 A118182 A107107

Adjacent sequences: A035095 A035096 A035097 this_sequence A035099 A035100 A035101

nonn

George Beck (beck(AT)wri.com)

More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Sep 23 2003