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Search: id:A141799
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| A141799 |
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Number of repeated integer partitions of n. An integer n can be partitioned into P(n) partitions P([n],i) where i=1,...,P(n) counts the partitions. The partition P([n],i) consists of T(n,i) integer parts t(i,j) with j=1,...,T(n,i). Now we perform on each t(i,j) an integer partition again and arrive at new partitions. Their parts can be partitioned again and so forth. We count the number of such repeated partitions of n. One convention is necessary to avoid an infinite loop: The trivial partition P([n],1)=[n] will not be partitioned again but just counted once (and therefore we also have a(1)=1). |
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| 1, 3, 8, 25, 66, 192, 511, 1418, 3812, 10383, 27958, 75758, 204215, 551821, 1488561, 4018722, 10842422, 29262357, 78955472, 213063551, 574905487, 1551325859, 4185959285, 11295211039, 30478118079, 82240300045, 221911189754
(list; graph; listen)
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OFFSET
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1,2
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LINKS
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Thomas Wieder, Home Page.
Thomas Wieder, (Old) Home Page.
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FORMULA
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Let sum_{i=1}^P(n) denote the sum over all integer partitions P([n],i) of n. Let sum_{j=1}^T(i,j) denote the sum over all parts of the i-th integer partition. Then we have the recursive formula 1 if t(i,j)=n a(n) = sum_{i=1}^P(n) sum_{j=1}^T(i,j) { a(t(i,j)) else. E.g. a(4)=25 because [4] contributes 1, [1,3] contributes a(1)+a(3)=1+8=9, [2,2] contributes a(2)+a(2)=3+3=6, [1,1,2] contributes a(1)+a(1)+a(2)=1+1+3=5, [1,1,1,1] contributes a(1)+a(1)+a(1)+a(1)=1+1+1+1=4 which gives in total 25.
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EXAMPLE
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For the integers 1, 2, 3 and 4 we have
[1] -> 1,
thus a(1)=1.
[2] -> 1,
[1,1] => [1] ->, [1] -> 1.
thus a(2)=3.
[3] -> 1,
[1,2] => [1] -> 1, [2] -> 3,
[1,1,1] => [1] -> 1, [1] -> 1, [1] -> 1,
thus a(3)=8.
[4] -> 1,
[1,3] => [1] -> 1, [3] -> 8,
[2,2] => [2] -> 3, [2] -> 3,
[1,1,2] => [1] -> 1, [1] -> 1, [2] -> 3,
[1,1,1,1] => [1] -> 1, [1] -> 1, [1] -> 1, [1] -> 1,
thus a(4)=25.
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MAPLE
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A141799 := proc(n) option remember ; local a, P, i, p ; if n =1 then 1; else a := 0 ; for P in combinat[partition](n) do if nops(P) > 1 then for i in P do a := a+procname(i) ; od: else a := a+1 ; fi; od: RETURN(a) ; fi ; end: for n from 1 to 40 do printf("%d, ", A141799(n)) ; od: [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 25 2008]
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CROSSREFS
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Cf. A000041, A131407, A131408, A137732.
Sequence in context: A026955 A093900 A018789 this_sequence A093969 A006177 A148788
Adjacent sequences: A141796 A141797 A141798 this_sequence A141800 A141801 A141802
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KEYWORD
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nonn
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AUTHOR
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Thomas Wieder (thomas.wieder(AT)t-online.de), Jul 05 2008
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EXTENSIONS
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Extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 25 2008
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