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A051292 Whitney number of level n of the lattice of the ideals of the crown of size 2 n. 9

%I

%S 2,1,1,4,9,21,52,127,313,778,1941,4863,12228,30837,77967,197574,

%T 501657,1275987,3250618,8292703,21182509,54169966,138674031,355343469,

%U 911347684,2339226871,6008781637,15445521202,39728258103,102248793573,263306364822,678411876729,1748800672089

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

%C 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

%C 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

%H Vincenzo Librandi, <a href="/A051292/b051292.txt">Table of n, a(n) for n = 0..1000</a>

%H Alessandro Conflitti, <a href="https://arxiv.org/abs/math/0505636">On Whitney numbers of the Order Ideals of Generalized Fences and Crowns</a>, arXiv:math/0505636 [math.CO], 2005.

%H E. Munarini, N. Zagaglia Salvi, <a href="http://dx.doi.org/10.1016/S0012-365X(02)00378-3">On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns</a>, Discrete Mathematics 259 (2002), 163-177.

%F 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).

%F 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

%F 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

%F 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

%F a(n) ~ 5^(1/4)*((1+sqrt(5))/2)^(2*n)/(2*sqrt(Pi*n)). - _Vaclav Kotesovec_, Jan 05 2013

%F 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

%e 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.

%p 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):

%p map(f, [$0..40]); # _Robert Israel_, Dec 06 2017

%p 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

%t 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 *)

%o (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

%Y Cf. A051291, A051286. Main diagonal of A205810.

%K nonn,easy

%O 0,1

%A _Emanuele Munarini_

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Last modified August 3 20:08 EDT 2020. Contains 336201 sequences. (Running on oeis4.)