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A051292 Whitney number of level n of the lattice of the ideals of the crown of size 2 n. 9
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, 263306364822, 678411876729, 1748800672089 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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, 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, May 07 2008

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.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Alessandro Conflitti, On Whitney numbers of the Order Ideals of Generalized Fences and Crowns

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, Jan 31 2005

Conjecture: n*(n-3)*a(n) -(2*n-1)*(n-3)*a(n-1) +(-n^2+4*n-5)*a(n-2) -(n-1)*(2*n-7)*a(n-3) +(n-1)*(n-4)*a(n-4)=0. - R. J. Mathar, Nov 30 2012

a(n) ~ 5^(1/4)*((1+sqrt(5))/2)^(2*n)/(2*sqrt(Pi*n)). - Vaclav Kotesovec, Jan 05 2013

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.

MATHEMATICA

CoefficientList[Series[(1-x^2+Sqrt[1-2*x-x^2-2*x^3+x^4])/Sqrt[1-2*x-x^2-2*x^3+x^4], {x, 0, 20}], x] (* Vaclav Kotesovec, Jan 05 2013 *)

PROG

(PARI) x='x+O('x^66); Vec( (1-x^2+sqrt(1-2*x-x^2-2*x^3+x^4))/sqrt(1-2*x-x^2-2*x^3+x^4) ) \\ Joerg Arndt, May 04 2013

CROSSREFS

Cf. A051291, A051286. main diagonal of A205810.

Sequence in context: A111569 A213786 A055130 * A094424 A265241 A166888

Adjacent sequences:  A051289 A051290 A051291 * A051293 A051294 A051295

KEYWORD

nonn,easy

AUTHOR

Emanuele Munarini

STATUS

approved

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Last modified March 28 22:27 EDT 2017. Contains 284249 sequences.