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
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.
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
EXAMPLE
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.
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 *)
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
KEYWORD
nonn,easy
AUTHOR
STATUS
approved