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A056823
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Number of compositions minus number of partitions: A011782(n) - A000041(n).
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27
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0, 0, 0, 1, 3, 9, 21, 49, 106, 226, 470, 968, 1971, 3995, 8057, 16208, 32537, 65239, 130687, 261654, 523661, 1047784, 2096150, 4193049, 8387033, 16775258, 33551996, 67105854, 134214010, 268430891, 536865308, 1073734982, 2147475299, 4294957153, 8589922282
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OFFSET
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0,5
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COMMENTS
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Previous name was: Counts members of A056808 by number of factors.
a(n) is also the number of compositions of n that are not partitions of n. - Omar E. Pol, Jan 31 2009, Oct 14 2013
a(n) is the number of compositions of n into positive parts containing pattern [1,2]. - Bob Selcoe, Jul 08 2014
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LINKS
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FORMULA
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G.f.: (1 - x) / (1 - 2*x) - Product_{k>=1} 1 / (1 - x^k). - Ilya Gutkovskiy, Jan 30 2020
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EXAMPLE
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A011782 begins 1 1 2 4 8 16 32 64 128 256 ...;
A000041 begins 1 1 2 3 5 7 11 15 22 30 ...;
so sequence begins 0 0 0 1 3 9 21 49 106 226 ... .
For n = 3 the factorizations are 8=2*2*2, 12=2*2*3, 18=2*3*3 and 30=2*3*5.
a(5) = 9: {[1,1,1,2], [1,1,2,1], [1,1,3], [1,2,1,1], [1,2,2], [1,3,1], [1,4], [2,1,2], [2,3]}. - Bob Selcoe, Jul 08 2014
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MAPLE
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a:= n-> ceil(2^(n-1))-combinat[numbpart](n):
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MATHEMATICA
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Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], !GreaterEqual@@#&]], {n, 0, 10}] (* Gus Wiseman, Jun 24 2020 *)
a[n_] := If[n == 0, 0, 2^(n-1) - PartitionsP[n]];
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CROSSREFS
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The version for patterns is A002051.
(1,2)-avoiding compositions are just partitions A000041.
The (1,1)-matching version is A261982.
The version for prime indices is A335447.
(1,2)-matching compositions are ranked by A335485.
Patterns matched by compositions are counted by A335456.
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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