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A185325
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Number of partitions of n into parts >= 5.
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21
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1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 10, 13, 15, 18, 21, 26, 30, 36, 42, 50, 58, 70, 80, 95, 110, 129, 150, 176, 202, 236, 272, 317, 364, 423, 484, 560, 643, 740, 847, 975, 1112, 1277, 1456, 1666, 1897, 2168, 2464, 2809, 3189, 3627, 4112, 4673
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OFFSET
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0,11
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COMMENTS
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a(n) is also the number of not necessarily connected 2-regular graphs on n-vertices with girth at least 5 (all such graphs are simple). The integer i corresponds to the i-cycle; addition of integers corresponds to disconnected union of cycles.
By removing a single part of size 5, an A026798 partition of n becomes an A185325 partition of n - 5. Hence this sequence is essentially the same as A026798.
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LINKS
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Table of n, a(n) for n=0..60.
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g
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FORMULA
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G.f.: Product 1/(1-x^m); m=5..inf.
Given by p(n)-p(n-1)-p(n-2)+2p(n-5)-p(n-8)-p(n-9)+p(n-10)where p(n)=A000041(n). [From Shanzhen Gao, Oct 28 2010; sign of 10 corrected from + to -, and moved from A026798 to this sequence by Jason Kimberley.]
This sequence is the Euler transformation of A185115.
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PROG
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(MAGMA)
p := func< n | n lt 0 select 0 else NumberOfPartitions(n) >;
A185325 := func<n | p(n)-p(n-1)-p(n-2)+2*p(n-5)-p(n-8)-p(n-9)+p(n-10)>;
[A185325(n):n in[0..60]];
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CROSSREFS
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2-regular simple graphs with girth at least 5: A185115 (connected), A185225 (disconnected), this sequence (not necessarily connected)
Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: A026807 (triangle); chosen g: A000041 (g=1 -- multigraphs with loops allowed), A002865 (g=2 -- multigraphs with loops forbidden), A008483 (g=3), A008484 (g=4), this sequence (g=5), A185326 (g=6), A185327 (g=7), A185328 (g=8), A185329 (g=9).
Not necessarily connected 2-regular graphs with girth exactly g [partitions with smallest part g]: A026794 (triangle); chosen g: A002865 (g=2), A026796 (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), A026800(g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10).
Not necessarily connected k-regular simple graphs with girth at least 5: A185315 (any k), A185305 (triangle); specified degree k: this sequence (k=2), A185335 (k=3).
Sequence in context: A096749 A036821 A026798 * A125890 A067661 A210024
Adjacent sequences: A185322 A185323 A185324 * A185326 A185327 A185328
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KEYWORD
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nonn,easy
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AUTHOR
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Jason Kimberley, Nov 11 2011
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STATUS
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approved
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