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A143141
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Total number of all repeated partitions of the integer n and its parts down to parts equal to 1. Essentially first differences of A055887.
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2
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 1..1000
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FORMULA
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a(n) = A055887(n) - A055887(n-1), n>1.
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EXAMPLE
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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).
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MAPLE
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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) ;
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MATHEMATICA
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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 *)
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CROSSREFS
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Cf. A143140, A055887, A000041, A141799, A131408, A137732.
Sequence in context: A355387 A077938 A077987 * A117294 A148306 A148307
Adjacent sequences: A143138 A143139 A143140 * A143142 A143143 A143144
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KEYWORD
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nonn
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AUTHOR
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Thomas Wieder, Jul 27 2008
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EXTENSIONS
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Edited and extended by R. J. Mathar, Aug 25 2008
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STATUS
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approved
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