A143141 Total number of all repeated partitions of the integer n and its parts down to parts equal to 1. Essentially first differences of A055887.

1, 2, 5, 14, 37, 101, 271, 733, 1976, 5334, 14390, 38833, 104779, 282734, 762903, 2058571, 5554692, 14988400, 40443620, 109130216, 294469216, 794574883, 2144024501, 5785283758, 15610599502, 42122535067, 113660462337, 306693333868, 827559549428, 2233028019698

Offset

1,2

Comments

Start from the A000041(n) integer partitions P(n,i,s) of the integer n at stage s=1.

The index i=1,...,A000041(n) denotes the different partitions.

We call the index s the partition stage and increase it by one as we sub-partition the partitions of a previous stage.

Each P(n,i,s) is a set P(n,i,s)={t(n,1,j,s)),...,t(P,i,j,s),...,t(P,i,J,s)} of parts t(P,i,j,s) of S.

The index j is attached to the parts of a partition P(n,i,s). 1<=j<=n since there are at most n parts.

Now apply the set partition process on every P(n,i,s=1).

That is, each t(n,i,j,s=1) is subjected to a further partitioning.

We get partitions P(t'(n,i,j,1),i',j',2)={t'(t(n,i,j,1),i',1,2),...,t'(t(n,i,j,1), i',j',2),...,t'(t(n,i,j,1),i',J',2)} of the second partition stage.

We repeat this partitioning process on each part t'(i,j',2) until we arrive at parts equal to 1 which cannot be partitioned any further.

We may speak of the full decomposition F of n into parts.

The sequence counts the total number of partitions of all stages of the full decomposition of n.

Note that n is its own partition, e.g. P(n=3,i=1,s=1)={3} is an integer partition of n=3.

We do not apply the repeated partitioning on the partition P(n,i,s)={n} (otherwise an infinite loop would arise).

For n=1 and n=2 there is no second partition stage: s stays at s=1.

The corresponding labeled counterpart is sequence A143140.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Formula

a(n) = A055887(n) - A055887(n-1), n>1.

Example

n=1:
[[1]]
n=2:
[[2], [1, 1]]
n=3:
[[3], [2, 1], [1, 1, 1]], [[2], [1, 1]]
n=4 in more detail:
[4], [3, 1], [2, 2], [2, 1, 1], [1, 1, 1, 1]], <- stage s=1, partition of 4
[[3], [2, 1], [1, 1, 1]], <- stage s=2 partitioning the first 3 of the 2nd partition
[[2], [1, 1]], <- stage s=2 partitioning the first 2 of the 3rd partition
[[2], [1, 1]], <- stage s=2 partitioning the second 2 of the 3rd partition
[[2], [1, 1]] <- stage s=2 partitioning the first 2 of the 4th partition
a(4) = 14 = 5 (from s=1)+9 (from s=2).

Maple

A055887 := proc(n) option remember ; if n = 0 then 1; else add(combinat[numbpart](k)*procname(n-k), k=1..n) ; fi; end: A143141 := proc(n) if n = 1 then 1; else A055887(n)-A055887(n-1) ; fi; end: seq(A143141(n), n=1..20) ;

Mathematica

b[n_] := b[n] = Sum[PartitionsP[k]*b[n-k], {k, 1, n}]; b[0]=1; A055887 = Table[b[n], {n, 0, 30}]; Join[{1}, Rest[Differences[A055887]]] (* Jean-François Alcover, Feb 05 2017 *)

Crossrefs

Cf. A143140, A055887, A000041, A141799, A131408, A137732.

Sequence in context: A355387 A077938 A077987 * A117294 A148306 A148307

Adjacent sequences: A143138 A143139 A143140 * A143142 A143143 A143144

Keyword

nonn

Author

Thomas Wieder, Jul 27 2008

Extensions

Edited and extended by R. J. Mathar, Aug 25 2008

Status

approved