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A026800
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Number of partitions of n in which the least part is 7.
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14
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0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 10, 11, 13, 15, 18, 20, 24, 27, 32, 36, 42, 48, 56, 63, 73, 83, 96, 108, 125, 141, 162, 183, 209, 236, 270, 304, 346, 390, 443, 498, 565, 635, 719, 807, 911, 1022, 1153, 1291, 1453, 1628, 1829, 2045
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,22
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COMMENTS
| Contribution by Jason Kimberley, Feb 03 2011: (Start)
a(n) is also the number of not necessarily connected 2-regular graphs on n-vertices with girth exactly 7 (all such graphs are simple). The integer i corresponds to the i-cycle; the addition of integers corresponds to the disconnected union of cycles.
By removing a single part of size 7, an A026800 partition of n becomes an A185327 partition of n - 7. (End)
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LINKS
| Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g
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FORMULA
| G.f.: x^7 * Product 1/(1-x^m); m=7..inf.
a(n) = p(n-7)-p(n-8)-p(n-9)+p(n-12)+2*p(n-14)-p(n-16)-p(n-17)-p(n-18)-p(n-19)+2*p(n-21)+p(n-23)-p(n-26)-p(n-27)+p(n-28) where p(n)=A000041(n) including the implicit p(n)=0 for negative n. [From Shanzhen Gao (shanzhengao(AT)yahoo.com), Oct 28 2010] - offset corrected / made explicit by Jason Kimberley, Feb 03 2011.
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EXAMPLE
| a(0)=0 because there does not exist a least part of the empty partition.
The a(7)=1 partition is 7.
The a(14)=1 partition is 7+7.
The a(15)=1 partition is 7+8.
.............................
The a(20)=1 partition is 7+13.
The a(21)=2 partitions are 7+7+7 and 7+14.
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PROG
| (MAGMA) p := func< n | n lt 0 select 0 else NumberOfPartitions(n) >;
A026800 := func< n | p(n-7)-p(n-8)-p(n-9)+p(n-12)+2*p(n-14)-p(n-16)- p(n-17)-p(n-18)-p(n-19)+2*p(n-21)+p(n-23)-p(n-26)-p(n-27)+p(n-28) >; // Jason Kimberley, Feb 03 2011
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CROSSREFS
| Cf. A185327 (Mathematica code)
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), A185325(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 -- multigraphs with at least one pair of parallel edges, but loops forbidden), A026796 (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), this sequence (g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10). - Jason Kimberley, Feb 03 2011
Sequence in context: A026825 A025150 * A185327 A171962 A029028 A029072
Adjacent sequences: A026797 A026798 A026799 * A026801 A026802 A026803
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KEYWORD
| nonn,easy,changed
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu)
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EXTENSIONS
| More terms from Arlin Anderson (starship1(AT)gmail.com), Apr 12 2001
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