%I A051286
%S A051286 1,1,2,5,11,26,63,153,376,931,2317,5794,14545,36631,92512,234205,
%T A051286 594169,1510192,3844787,9802895,25027296,63972861,163701327,419316330,
%U A051286 1075049011,2758543201,7083830648,18204064403,46812088751,120452857976
%N A051286 Whitney number of level n of the lattice of the ideals of the fence of 
               order 2 n.
%C A051286 A Chebyshev transform of the central trinomial numbers A002426: image 
               of 1/sqrt(1-2x-3x^2) under the mapping that takes g(x) to (1/(1+x^2))g(x/
               (1+x^2)). - Paul Barry (pbarry(AT)wit.ie), Jan 31 2005
%C A051286 a(n) has same parity as Fibonacci(n+1) = A000045(n+1); see A107597. - 
               Paul D. Hanna (pauldhanna(AT)juno.com), May 22 2005
%C A051286 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
%D A051286 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.
%F A051286 G.f.: 1/sqrt(1-2*t-t^2-2*t^3+t^4).
%F A051286 a(n)=sum{k=0..floor(n/2), binomial(n-k, k)(-1)^k*A002426(n-2k)}. - Paul 
               Barry (pbarry(AT)wit.ie), Jan 31 2005
%F A051286 a(n) = Sum_{k=0..n} C(n-k, k)^2. Limit_{n->inf} a(n+1)/a(n) = (sqrt(5)+3)/
               2. G.f.: A(x) = 1/sqrt((1+x+x^2)*(1-3*x+x^2)). - Paul D. Hanna (pauldhanna(AT)juno.com), 
               May 22 2005
%F A051286 a(n) = Sum_{k=0..n} A049310(n, k)^2 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Nov 21 2005
%e A051286 a(3) = 5 because the ideals of size 3 of the fence F(6) = { x1 < x2 > 
               x3 < x4 > x5 < x6 } are x1x3x5, x1x2x3, x3x4x5, x1x5x6, x3x5x6.
%p A051286 seq( sum('binomial(i-k,k)*binomial(i-k,k)', 'k'=0..floor(i/2)), i=0..30 
               ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 09 2001
%o A051286 (PARI) a(n)=polcoeff(1/sqrt((1+x+x^2)*(1-3*x+x^2)+x*O(x^n)),n) (PARI) 
               a(n)=sum(k=0,n,binomial(n-k,k)^2) (Hanna)
%Y A051286 Cf. A051291, A051292.
%Y A051286 Cf. A107597.
%Y A051286 Sequence in context: A124217 A095981 A082397 this_sequence A025245 A079223 
               A095892
%Y A051286 Adjacent sequences: A051283 A051284 A051285 this_sequence A051287 A051288 
               A051289
%K A051286 nonn
%O A051286 0,3
%A A051286 Emanuele Munarini (munarini(AT)mate.polimi.it)