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A055887
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Number of ordered partitions of partitions.
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9
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1, 1, 3, 8, 22, 59, 160, 431, 1164, 3140, 8474, 22864, 61697, 166476, 449210, 1212113, 3270684, 8825376, 23813776, 64257396, 173387612, 467856828, 1262431711, 3406456212, 9191739970, 24802339472, 66924874539
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Jordan matrices are upper bidiagonal matrices such that (A) the diagonal entries are in sorted order, (B) there are only 1's and 0's on the superdiagonal, (C) for each superdiagonal 1, the two diagonal entries to the left and below it must be equal. Let J(N) be the number of N X N Jordan matrices where the diagonal values are, wlog, taken to be a prefix of some fixed strictly increasing sequence x_1, x_2, x_3, ... If Jordan blocks sorted by eigenvalue with ties broken by block size during the sorting, then J(1, 2, 3, ...) is this sequence. - Warren Smith, Jan 28 2002
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LINKS
| N. J. A. Sloane, Transforms
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.
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FORMULA
| Invert transform of partitions numbers A000041.
Let p(k) be the number of integer partitions of k. Furthermore, set a(0)=1. Then a(n) = Sum_{k=1..n} p(k)*a(n-k). - Thomas Wieder (thomas.wieder(AT)t-online.de), Nov 26 2007
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MAPLE
| with(combstruct); SeqSetSetU := [T, {T=Sequence(S), S=Set(U, card >= 1), U=Set(Z, card >=1)}, unlabeled];
P := (x) -> product( 1/(1-x^k), k=1..20 ) - 1; F := (x) -> series( 1/(1-P(x)) - 1, x, 21 ); F(x) is g.f. for this sequence - Warren Smith, Jan 28 2002
A055887rec := proc(n::integer) #with(combinat): local k; option remember; if n = 0 then 1 else add(numbpart(k) * procname(n - k), k=1..n); end if; end proc; - Thomas Wieder (thomas.wieder(AT)t-online.de), Nov 26 2007
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MATHEMATICA
| a = 1/Product[(1 - x^k), {k, 1, \[Infinity]}] - 1; CoefficientList[Series[1/(1 - a), {x, 0, 20}], x] (* From Geoffrey Critzer, Dec 23 2010 *)
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CROSSREFS
| Cf. A055888, A083355.
Sequence in context: A001853 A003227 A077848 * A024581 A028859 A155020
Adjacent sequences: A055884 A055885 A055886 * A055888 A055889 A055890
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KEYWORD
| nonn
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AUTHOR
| Christian G. Bower (bowerc(AT)usa.net), Jun 09 2000
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