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A137278
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Triangle read by rows: g(n,k) = number of ideals of size k in a garland (or double fence) of order n.
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3
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1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 3, 3, 3, 3, 1, 1, 4, 6, 6, 7, 6, 6, 4, 1, 1, 5, 10, 12, 14, 15, 14, 12, 10, 5, 1, 1, 6, 15, 22, 27, 32, 33, 32, 27, 22, 15, 6, 1, 1, 7, 21, 37, 50, 63, 72, 75, 72, 63, 50, 37, 21, 7, 1, 1, 8, 28, 58, 88, 118, 146, 164, 171, 164, 146, 118, 88, 58, 28, 8, 1
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,6
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COMMENTS
| Row n has 2n+1 terms.
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REFERENCES
| T. S. Blyth and J. C. Varlet, Ockham Algebras, Oxford Science Pub. 1994.
E. Munarini, Enumeration of order ideals of a garland, Ars Combin. 76 (2005), 185-192.
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LINKS
| Emanuele Munarini (emanuele.munarini(AT)polimi.it), Mar 13 2008, Table of n, a(n) for n = 0..440 [Rows 0 through 20, flattened]
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FORMULA
| G.f.: G(x,t) = {1-x^2t^2}/{1-(1+x+x^2)t+x^2t^2+x^3t^3}. Recurrence: g(n+3,k+3) = g(n+2,k+3) + g(n+2,k+2) + g(n+2,k+1) - g(n+1,k+1) - g(n,k)
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EXAMPLE
| In the garland
5..6..7..8
o..o..o..o
|\/|\/|\/|
|/\|/\|/\|
o..o..o..o
1..2..3..4
the ideals of size 4 are 1234, 1253, 1254, 1236, 2347, 1348, 2348.
Triangle begins:
1,
1, 1, 1,
1, 2, 1, 2, 1,
1, 3, 3, 3, 3, 3, 1,
1, 4, 6, 6, 7, 6, 6, 4, 1,
1, 5, 10, 12, 14, 15, 14, 12, 10, 5, 1,
1, 6, 15, 22, 27, 32, 33, 32, 27, 22, 15, 6, 1,
...
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CROSSREFS
| Sequence in context: A131332 A059780 A075119 * A205810 A139368 A134303
Adjacent sequences: A137275 A137276 A137277 * A137279 A137280 A137281
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KEYWORD
| easy,nonn,tabf
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AUTHOR
| Emanuele Munarini (emanuele.munarini(AT)polimi.it), Mar 13 2008
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