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A052703 A simple context-free grammar: convolution cube of A001002.

%I A052703 #30 Jan 13 2025 10:08:16
%S A052703 0,0,0,1,3,12,49,210,927,4191,19305,90285,427570,2046324,9881862,
%T A052703 48090824,235619133,1161257580,5753365015,28638093270,143148398085,
%U A052703 718242481770,3616135914375,18263111515740,92500790125770,469737499557222,2391192703656054,12199557377107450
%N A052703 A simple context-free grammar: convolution cube of A001002.
%H A052703 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=656">Encyclopedia of Combinatorial Structures 656</a>
%F A052703 G.f.: RootOf(-_Z+_Z^2+_Z^3+x)-RootOf(-_Z+_Z^2+_Z^3+x)^2-x
%F A052703 Recurrence: {a(1)=0, a(2)=0, a(3)=1, a(4)=3, (30-135*n+135*n^2)*a(n)+(-130-107*n+29*n^2)*a(n+1)+(-281*n-198-91*n^2)*a(n+2)+(15*n^2+75*n+90)*a(n+3)}
%F A052703 From _Seiichi Manyama_, Nov 22 2024: (Start)
%F A052703 G.f.: (x*B(x))^3 where B(x) is the g.f. of A001002.
%F A052703 a(n) = 3 * Sum_{k=0..n-3} binomial(n+k,k) * binomial(k,n-3-k)/(n+k). (End)
%F A052703 a(n) ~ 3^(3*n - 5/2) / (sqrt(Pi) * 2^(3/2) * n^(3/2) * 5^(n - 1/2)). - _Vaclav Kotesovec_, Nov 22 2024
%p A052703 spec := [S,{C=Prod(B,B),B=Union(S,C,Z),S=Prod(B,C)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
%o A052703 (PARI) my(N=30, x='x+O('x^N)); concat([0, 0, 0], Vec(serreverse(x-x^2-x^3)-serreverse(x-x^2-x^3)^2-x)) \\ _Seiichi Manyama_, Nov 22 2024
%o A052703 (PARI) a(n) = 3*sum(k=0, n-3, binomial(n+k, k)*binomial(k, n-3-k)/(n+k)); \\ _Seiichi Manyama_, Nov 22 2024
%Y A052703 Cf. A001002, A052706,
%K A052703 easy,nonn
%O A052703 0,5
%A A052703 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052703 More terms from _Seiichi Manyama_, Nov 21 2024