OFFSET
0,5
COMMENTS
a(n)=Sum(k*A182896(n,k), k>=0).
REFERENCES
M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
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.: G(z)=z^3*c/[(1+z+z^2)(1-3z+z^2)], where c satisfies c = 1+zc+z^2*c+z^3*c^2.
a(n) ~ ((1 + sqrt(5))/2)^(2*n+1) / (4*sqrt(5)). - Vaclav Kotesovec, Mar 06 2016
D-finite with recurrence n*a(n) +(-4*n+3)*a(n-1) +(2*n-3)*a(n-2) +11*(n-3)*a(n-4) +(2*n-9)*a(n-6) +(-4*n+21)*a(n-7) +(n-6)*a(n-8)=0. - R. J. Mathar, Jul 24 2022
EXAMPLE
a(3)=1 because, denoting by h (H) the (1,0)-step of weight 1 (2), and u=(1,1), d=(1,-1), the five paths of weight 3 are ud, du, hH, Hh, and hhh; they contain 1+0+0+0+0=1 (1,-1)-return to the horizontal axis.
MAPLE
eq := c = 1+z*c+z^2*c+z^3*c^2: c := RootOf(eq, c): G := z^3*c/((1+z+z^2)*(1-3*z+z^2)): Gser := series(G, z = 0, 32): seq(coeff(Gser, z, n), n = 0 .. 29);
MATHEMATICA
CoefficientList[Series[x^3*(1 - x - x^2 - Sqrt[1+x^4-2*x^3-x^2-2*x]) / (2*x^3*(1+x+x^2)*(1-3*x+x^2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 06 2016 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Dec 13 2010
STATUS
approved