A052702 A simple context-free grammar.

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

0,6

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

Table of n, a(n) for n=0..38.

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

Sequence in context: A290991 A007910 A293315 * A058766 A127601 A030038

Adjacent sequences: A052699 A052700 A052701 * A052703 A052704 A052705

Keyword

easy,nonn

Author

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

Extensions

More terms from Joerg Arndt, May 11 2012

Status

approved