0,3

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

a(n) has same parity as Fibonacci(n+1) = A000045(n+1); see A107597. - Paul D. Hanna (pauldhanna(AT)juno.com), May 22 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 (zaslav(AT)math.binghamton.edu), May 07 2008

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.

G.f.: 1/sqrt(1-2*t-t^2-2*t^3+t^4).

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

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

a(n) = Sum_{k=0..n} A049310(n, k)^2 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 21 2005

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.

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

(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)

Cf. A051291, A051292.

Cf. A107597.

Sequence in context: A124217 A095981 A082397 this_sequence A025245 A079223 A095892

Adjacent sequences: A051283 A051284 A051285 this_sequence A051287 A051288 A051289

nonn

Emanuele Munarini (munarini(AT)mate.polimi.it)