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A051292
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Whitney number of level n of the lattice of the ideals of the crown of size 2 n.
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8
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2, 1, 1, 4, 9, 21, 52, 127, 313, 778, 1941, 4863, 12228, 30837, 77967, 197574, 501657, 1275987, 3250618, 8292703, 21182509, 54169966, 138674031, 355343469, 911347684, 2339226871, 6008781637, 15445521202, 39728258103, 102248793573
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| A Chebyshev transform of the central binomial numbers A002426 under the mapping that takes g(x) to ((1-x^2)/(1+x^2))g(x/(1+x^2)). Starts 1,1,1,4,9,21,... - Paul Barry (pbarry(AT)wit.ie), Jan 31 2005
This is the second kind of Whitney numbers, which count elements, not to be confused with the first kind, which sum Mobius functions. - Thomas Zaslavsky (zaslav(AT)math.binghamton.edu), May 07 2008
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REFERENCES
| E. Munarini and N. Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.
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LINKS
| Alessandro Conflitti, On Whitney numbers of the Order Ideals of Generalized Fences and Crowns
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FORMULA
| G.f.: (1-t^2+sqrt(1-2*t-t^2-2*t^3+t^4))/sqrt(1-2*t-t^2-2*t^3+t^4)
a(n)=sum{k=0..floor(n/2), (n/(n-k))C(n-k, k)*(-1)^k*sum{i=0..floor((n-2k)/2), C(n-2k, 2i)C(2i, i)}}; a(n)=sum{k=0..floor(n/2), (n/(n-k))C(n-k, k)*(-1)^k*A002426(n-2k)}. - Paul Barry (pbarry(AT)wit.ie), Jan 31 2005
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EXAMPLE
| a(3) = 4 because the ideals of size 3 of the crown C(3) = { x1 < x2 > x3 < x4 > x5 < x6 > x1 } are x1x2x3, x3x4x5, x1x6x5, x1x3x5.
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CROSSREFS
| Cf. A051291, A051286. main diagonal of A205810.
Sequence in context: A096540 A111569 A055130 * A094424 A166888 A083677
Adjacent sequences: A051289 A051290 A051291 * A051293 A051294 A051295
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KEYWORD
| nonn
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AUTHOR
| Emanuele Munarini (munarini(AT)mate.polimi.it)
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EXTENSIONS
| ArXiv URL replaced by non-cached version - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 23 2009
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