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A052706 A simple context-free grammar: convolution square of A001002.

%I A052706 #37 Jan 13 2025 10:05:19
%S A052706 0,0,1,2,7,26,105,444,1944,8734,40040,186550,880750,4204508,20260498,
%T A052706 98419392,481442805,2369551218,11725590555,58303117680,291151523355,
%U A052706 1459590130350,7342906908645,37058911816680,187579329483780,952006706210196,4843566974043900
%N A052706 A simple context-free grammar: convolution square of A001002.
%H A052706 Vincenzo Librandi, <a href="/A052706/b052706.txt">Table of n, a(n) for n = 0..200</a>
%H A052706 Nancy S. S. Gu, Nelson Y. Li, and Toufik Mansour, <a href="http://dx.doi.org/10.1016/j.disc.2007.04.007">2-Binary trees: bijections and related issues</a>, Discr. Math., 308 (2008), 1209-1221.
%H A052706 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=661">Encyclopedia of Combinatorial Structures 661</a>
%F A052706 G.f.: RootOf(-_Z+_Z^2+_Z^3+x)^2.
%F A052706 Recurrence: {a(1) = 0, a(2) = 1, a(3) = 2, (6-27*n+27*n^2)*a(n)+(6+65*n+49*n^2)*a(n+1)+(67*n+66+17*n^2)*a(n+2)+(-5*n^2-25*n-30)*a(n+3)}.
%F A052706 a(n) = 2*(Sum_{k=0..n-2} binomial(k,n-k-2)*binomial(n+k-1,n-1))/n, n>1, a(0)=a(1)=0. - _Vladimir Kruchinin_, May 19 2012
%F A052706 a(n) ~ 3^(3*n-5/2)/(sqrt(2*Pi)*5^(n-1/2)*n^(3/2)). - _Vaclav Kotesovec_, Oct 09 2012
%p A052706 spec := [S,{C = Union(S,B,Z),B = Prod(S,C),S = Prod(C,C)},unlabeled]: seq(combstruct[count](spec,size = n), n = 0..20);
%t A052706 Flatten[{0,0,Table[2*Sum[Binomial[k,n-k-2]*Binomial[n+k-1,n-1],{k,0,n-2}]/n,{n,2,20}]}] (* _Vaclav Kotesovec_, Oct 09 2012 *)
%o A052706 (Maxima) a(n):=if n<2 then 0 else (2*sum(binomial(k,n-k-2)*binomial(n+k-1,n-1),k,0,n-2))/n; /* _Vladimir Kruchinin_, May 19 2012 */
%o A052706 (PARI) a(n) = if(n>1,2*sum(k=0,n-2,binomial(k,n-k-2)*binomial(n+k-1,n-1))/n,0) \\ _Jason Yuen_, Aug 12 2024
%Y A052706 Cf. A001002.
%K A052706 easy,nonn
%O A052706 0,4
%A A052706 encyclopedia(AT)pommard.inria.fr, Jan 25 2000