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

0, 0, 1, 2, 7, 26, 105, 444, 1944, 8734, 40040, 186550, 880750, 4204508, 20260498, 98419392, 481442805, 2369551218, 11725590555, 58303117680, 291151523355, 1459590130350, 7342906908645, 37058911816680, 187579329483780, 952006706210196, 4843566974043900

Offset: 0

Author

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

Keywords

• easy
• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Nancy S. S. Gu, Nelson Y. Li, and Toufik Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 661

Crossrefs

Cf. A001002.

Programs

Maple

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);

Mathematica

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 *)

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 */

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

Formula

G.f.: RootOf(-_Z+_Z^2+_Z^3+x)^2.

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)}.

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

a(n) ~ 3^(3*n-5/2)/(sqrt(2*Pi)*5^(n-1/2)*n^(3/2)). - Vaclav Kotesovec, Oct 09 2012