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A052706
A simple context-free grammar.
2
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
0,4
LINKS
Nancy S. S. Gu, Nelson Y. Li, and Toufik Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.
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
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 *)
PROG
(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
CROSSREFS
Sequence in context: A150554 A150555 A151297 * A150556 A150557 A150558
KEYWORD
easy,nonn
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
STATUS
approved