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A136029
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a(n) is the number of central ideals of a garland of order 2n, i.e. a(n) = g(2n,n), where g(n,k) is the number of ideals of size k in a garland (or double fence) of order n (see A137278).
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3
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1, 1, 1, 3, 7, 15, 33, 75, 171, 391, 899, 2077, 4815, 11195, 26097, 60975, 142751, 334791, 786419, 1849905, 4357121, 10274313, 24252923, 57305241, 135521807, 320758587, 759757139, 1800838381, 4271267043, 10136815015, 24070870545
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Also the number of one-sided n-step prudent walks, starting from (0,0) and ending on the y-axis, with east, west and north steps. - Shanzhen Gao, Apr 26 2011
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REFERENCES
| T. S. Blyth, J. C. Varlet, Ockham algebras, Oxford Science Pub. 1994.
E. Munarini, Enumeration of order ideals of a garland, Ars Combin. 76 (2005), 185--192.
Munarini, Emanuele, Combinatorial properties of the antichains of a garland. Integers, 9 (2009), 353-374.
S. Gao, H. Niederhausen, Sequences Arising From Prudent Self-Avoiding Walks (submitted to INTEGERS: The Electronic Journal of Combinatorial Number Theory).
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LINKS
| Emanuele Munarini (emanuele.munarini(AT)polimi.it), Mar 21 2008, Table of n, a(n) for n = 0..100
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FORMULA
| Recurrence: (n+6)*a(n+6) - (2*n+11)*a(n+5) - (n+3)*a(n+4) - 4*a(n+3) - (n+4)*c_(n+2) - (2*n+3)*a(n+1) + (n+1), a(n) = 0.
G.f.: (1 - x^2)/sqrt( 1 - 2*x - x^2 - x^4 + 2*x^5 + x^6 ).
a(n)=1+sum(k=1..floor((n-1)/2), sum(i=1..min(n-2*k,k), binomial(n-2*k+1,i) * binomial(k-1,k-i) * binomial(n-k-i,k) ) ) [Shanzhen Gao, May 13 2011]
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EXAMPLE
| a(4) = 7, since the central ideals of the garland G(4):
5..6..7..8
o..o..o..o
|\/|\/|\/|
|/\|/\|/\|
o..o..o..o
1..2..3..4
are: 1234, 1253, 1254, 1236, 2347, 1348, 2348.
a(4)=7, since there are 7 such walks: NNNN, NENW, NWNE, ENWN, ENNW, WNEN, WNNE. [Shanzhen Gao, May 13 2011]
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CROSSREFS
| Sequence in context: A193641 A026701 A140498 * A101892 A199882 A147102
Adjacent sequences: A136026 A136027 A136028 * A136030 A136031 A136032
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KEYWORD
| easy,nonn
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AUTHOR
| Emanuele Munarini (emanuele.munarini(AT)polimi.it), Mar 21 2008
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