A052702 Expansion of (1/2)*(1/x^2 - 1/x)*(1-x-sqrt(1-2*x+x^2-4*x^3)) - x.

0, 0, 0, 0, 1, 2, 3, 6, 13, 26, 52, 108, 226, 472, 993, 2106, 4485, 9586, 20576, 44332, 95814, 207688, 451438, 983736, 2148618, 4702976, 10314672, 22664452, 49887084, 109985772, 242854669, 537004218, 1189032613, 2636096922, 5851266616, 13002628132, 28925389870, 64412505472, 143576017410

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Comments

From Paul Barry, May 24 2009: (Start)

Hankel transform of A052702 is A160705. Hankel transform of A052702(n+1) is A160706.

Hankel transform of A052702(n+2) is -A131531(n+1). Hankel transform of A052702(n+3) is A160706(n+5).

Hankel transform of A052702(n+4) is A160705(n+5). (End)

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

Crossrefs

Cf. A023431.

Programs

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 *)

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 */

Formula

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)

a(n) = A023431(n-1)-A023431(n-2). - R. J. Mathar, Jan 13 2025

Extensions

More terms from Joerg Arndt, May 11 2012