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%I #33 Mar 04 2024 01:39:37
%S 0,0,0,0,1,2,3,6,13,26,52,108,226,472,993,2106,4485,9586,20576,44332,
%T 95814,207688,451438,983736,2148618,4702976,10314672,22664452,
%U 49887084,109985772,242854669,537004218,1189032613,2636096922,5851266616,13002628132,28925389870,64412505472,143576017410
%N A simple context-free grammar.
%C From _Paul Barry_, May 24 2009: (Start)
%C Hankel transform of A052702 is A160705. Hankel transform of A052702(n+1) is A160706.
%C Hankel transform of A052702(n+2) is -A131531(n+1). Hankel transform of A052702(n+3) is A160706(n+5).
%C Hankel transform of A052702(n+4) is A160705(n+5). (End)
%C 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
%H Ricardo Gómez Aíza, <a href="https://arxiv.org/abs/2402.16111">Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis</a>, arXiv:2402.16111 [math.CO], 2024. See pp. 10, 19-21.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=654">Encyclopedia of Combinatorial Structures 654</a>
%F G.f.: (1/2)*(1/x^2 - 1/x)*(1-x-sqrt(1-2*x+x^2-4*x^3)) - x.
%F 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)}.
%F From _Paul Barry_, May 24 2009: (Start)
%F G.f.: (1-2*x+x^2-2*x^3-(1-x)*sqrt(1-2*x+x^2-4*x^3))/(2*x^2).
%F a(n+1) = Sum_{k=0..n-1} C(n-k-1,2k-1)*A000108(k). (End)
%p 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);
%t a[n_] := Sum[Binomial[n-k-2, 2k-1] CatalanNumber[k], {k, 0, n-2}];
%t Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Oct 11 2022, after _Paul Barry_ *)
%o (PARI)
%o x='x+O('x^66);
%o s='a0+(1-2*x+x^2-2*x^3-(1-x)*sqrt(1-2*x+x^2-4*x^3))/(2*x^2);
%o v=Vec(s); v[1]-='a0; v
%o /* _Joerg Arndt_, May 11 2012 */
%K easy,nonn
%O 0,6
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E More terms from _Joerg Arndt_, May 11 2012
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