OFFSET
0,6
COMMENTS
From Paul Barry, May 24 2009: (Start)
Hankel transform of A052702(n+2) is -A131531(n+1). Hankel transform of A052702(n+3) is A160706(n+5).
For n > 1, number of Dyck (n-1)-paths with each descent length one greater or one less than the preceding ascent length. - David Scambler, May 11 2012
LINKS
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See pp. 10, 19-21.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 654
FORMULA
G.f.: (1/2)*(1/x^2 - 1/x)*(1-x-sqrt(1-2*x+x^2-4*x^3)) - x.
Recurrence: {a(1)=0, a(2)=0, a(4)=1, a(3)=0, a(6)=3, a(7)=6, a(5)=2, (-2+4*n)*a(n)+(-7-5*n)*a(n+1)+(8+3*n)*a(n+2)+(-13-3*n)*a(n+3)+(n+6)*a(n+4)}.
From Paul Barry, May 24 2009: (Start)
G.f.: (1-2*x+x^2-2*x^3-(1-x)*sqrt(1-2*x+x^2-4*x^3))/(2*x^2).
a(n+1) = Sum_{k=0..n-1} C(n-k-1,2k-1)*A000108(k). (End)
MAPLE
spec := [S, {B=Prod(C, Z), S=Prod(B, B), C=Union(S, B, Z)}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
MATHEMATICA
a[n_] := Sum[Binomial[n-k-2, 2k-1] CatalanNumber[k], {k, 0, n-2}];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Oct 11 2022, after Paul Barry *)
PROG
(PARI)
x='x+O('x^66);
s='a0+(1-2*x+x^2-2*x^3-(1-x)*sqrt(1-2*x+x^2-4*x^3))/(2*x^2);
v=Vec(s); v[1]-='a0; v
/* Joerg Arndt, May 11 2012 */
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
EXTENSIONS
More terms from Joerg Arndt, May 11 2012
STATUS
approved