A051292 Whitney number of level n of the lattice of the ideals of the crown of size 2 n.

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

Offset: 0

Author

Emanuele Munarini

Keywords

• nonn
• easy

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

Examples

			a(3) = 4 because the ideals of size 3 of the crown C(3) = { x1 < x2 > x3 < x4 > x5 < x6 > x1 } are x1*x2*x3, x3*x4*x5, x1*x6*x5, x1*x3*x5.
		

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, arXiv:math/0505636 [math.CO], 2005.
• E. Munarini, N. Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.

Crossrefs

Cf. A051291, A051286. Main diagonal of A205810.

Programs

Maple

f:= gfun:-rectoproc({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, a(0) = 2, a(1) = 1, a(2) = 1, a(3) = 4},a(n),remember):
map(f, [$0..40]); # Robert Israel, Dec 06 2017
a := n -> `if`(n=0,2,2*add(((1+(-1)^(n-k)))*n*k*binomial((n+k)/2, k)^2*1/((n+k))^2, k=0..n)): seq(a(n), n=0..32); # Leonid Bedratyuk, Dec 07 2017

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 *)

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

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

Conjecture confirmed using the differential equation (2*x^2-x+2)*y(x) + (4*x^4-5*x^3-x^2+x-2)*y'(x) + (x^5-2*x^4-x^3-2*x^2+x)*y''(x) - 2*x^2 + x - 2 = 0 satisfied by the g.f. - Robert Israel, Dec 06 2017

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

a(n) = 2n Sum_{k=0..n}(1+(-1)^(n-k))*C((n+k)/2,k)^2*k/((n+k))^2 for n > 0. - Leonid Bedratyuk, Dec 06 2017