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A052702 A simple context-free grammar. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

Contribution 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.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 654

FORMULA

G.f.: (1/2)/x^2*(1-x-(1-2*x+x^2-4*x^3)^(1/2))-(1/2)/x*(1-x-(1-2*x+x^2-4*x^3)^(1/2))-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)}.

Contribution 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);

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

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Last modified October 28 13:26 EDT 2020. Contains 338055 sequences. (Running on oeis4.)