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%I #12 Feb 05 2017 09:31:07
%S 1,2,5,14,37,101,271,733,1976,5334,14390,38833,104779,282734,762903,
%T 2058571,5554692,14988400,40443620,109130216,294469216,794574883,
%U 2144024501,5785283758,15610599502,42122535067,113660462337,306693333868,827559549428,2233028019698
%N Total number of all repeated partitions of the integer n and its parts down to parts equal to 1. Essentially first differences of A055887.
%C Start from the A000041(n) integer partitions P(n,i,s) of the integer n at stage s=1.
%C The index i=1,...,A000041(n) denotes the different partitions.
%C We call the index s the partition stage and increase it by one as we sub-partition the partitions of a previous stage.
%C 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.
%C 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.
%C Now apply the set partition process on every P(n,i,s=1).
%C That is, each t(n,i,j,s=1) is subjected to a further partitioning.
%C 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.
%C 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.
%C We may speak of the full decomposition F of n into parts.
%C The sequence counts the total number of partitions of all stages of the full decomposition of n.
%C Note that n is its own partition, e.g. P(n=3,i=1,s=1)={3} is an integer partition of n=3.
%C We do not apply the repeated partitioning on the partition P(n,i,s)={n} (otherwise an infinite loop would arise).
%C For n=1 and n=2 there is no second partition stage: s stays at s=1.
%C The corresponding labeled counterpart is sequence A143140.
%H Alois P. Heinz, <a href="/A143141/b143141.txt">Table of n, a(n) for n = 1..1000</a>
%F a(n) = A055887(n) - A055887(n-1), n>1.
%e n=1:
%e [[1]]
%e n=2:
%e [[2], [1, 1]]
%e n=3:
%e [[3], [2, 1], [1, 1, 1]], [[2], [1, 1]]
%e n=4 in more detail:
%e [4], [3, 1], [2, 2], [2, 1, 1], [1, 1, 1, 1]], <- stage s=1, partition of 4
%e [[3], [2, 1], [1, 1, 1]], <- stage s=2 partitioning the first 3 of the 2nd partition
%e [[2], [1, 1]], <- stage s=2 partitioning the first 2 of the 3rd partition
%e [[2], [1, 1]], <- stage s=2 partitioning the second 2 of the 3rd partition
%e [[2], [1, 1]] <- stage s=2 partitioning the first 2 of the 4th partition
%e a(4) = 14 = 5 (from s=1)+9 (from s=2).
%p 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) ;
%t 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 *)
%Y Cf. A143140, A055887, A000041, A141799, A131408, A137732.
%K nonn
%O 1,2
%A _Thomas Wieder_, Jul 27 2008
%E Edited and extended by _R. J. Mathar_, Aug 25 2008
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