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A098859
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Number of partitions of n into parts each of which is used a different number of times.
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13
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1, 1, 2, 2, 4, 5, 7, 10, 13, 15, 21, 28, 31, 45, 55, 62, 82, 105, 116, 153, 172, 208, 251, 312, 341, 431, 492, 588, 676, 826, 905, 1120, 1249, 1475, 1676, 2003, 2187, 2625, 2922, 3409, 3810, 4481, 4910, 5792, 6382, 7407, 8186, 9527, 10434
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OFFSET
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0,3
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COMMENTS
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Fill, Janson and Ward refer to these partitions as Wilf partitions. - Peter Luschny, Jun 04 2012
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LINKS
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Maciej Ireneusz Wilczynski, Table of n, a(n) for n = 0..700
James Allen Fill, Svante Janson and Mark Daniel Ward, Partitions with Distinct Multiplicities of Parts: On An "Unsolved Problem" Posed By Herbert Wilf, The Electronic Journal of Combinatorics, Volume 19, Issue 2 (2012)
Daniel Kane and Robert C. Rhoades, Asymptotics for Wilf's partitions with distinct multiplicities
Doron Zeilberger, Using generatingfunctionology to enumerate distinct-multiplicity partitions.
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FORMULA
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log(a(n)) ~ N*log(N) where N = (6*n)^(1/3) (see Fill, Janson and Ward). - Peter Luschny, Jun 04 2012
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EXAMPLE
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a(6)=7 because 6= 4+1+1= 3+3= 3+1+1+1= 2+2+2= 2+1+1+1+1= 1+1+1+1+1+1. Four unrestricted partitions of 6 are not counted by a(6): 5+1, 4+2, 3+2+1 because at least two different summands are each used once; 2+2+1+1 because each summand is used twice.
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MATHEMATICA
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a[n_] := Length[sp = Split /@ IntegerPartitions[n]] - Count[sp, {___List, b_List, ___List, c_List, ___List} /; Length[b] == Length[c]]; Table[a[n], {n, 0, 48}] (* Jean-François Alcover, Jan 17 2013 *)
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PROG
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(Haskell)
a098859 = p 0 [] 1 where
p m ms _ 0 = if m `elem` ms then 0 else 1
p m ms k x
| x < k = 0
| m == 0 = p 1 ms k (x - k) + p 0 ms (k + 1) x
| m `elem` ms = p (m + 1) ms k (x - k)
| otherwise = p (m + 1) ms k (x - k) + p 0 (m : ms) (k + 1) x
-- Reinhard Zumkeller, Dec 27 2012
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CROSSREFS
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Row sums of A182485.
Cf. A100471, A100881, A105637, A211858, A211859, A211860, A211861, A211862, A211863.
Sequence in context: A026930 A211862 A211863 * A195012 A034398 A027069
Adjacent sequences: A098856 A098857 A098858 * A098860 A098861 A098862
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KEYWORD
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nonn,nice,changed
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AUTHOR
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David S. Newman, Oct 11 2004
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EXTENSIONS
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Corrected and extended by Vladeta Jovovic, Oct 22 2004
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STATUS
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approved
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