A052703 A simple context-free grammar: convolution cube of A001002.
0, 0, 0, 1, 3, 12, 49, 210, 927, 4191, 19305, 90285, 427570, 2046324, 9881862, 48090824, 235619133, 1161257580, 5753365015, 28638093270, 143148398085, 718242481770, 3616135914375, 18263111515740, 92500790125770, 469737499557222, 2391192703656054, 12199557377107450
Offset: 0
Links
- INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 656
Programs
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Maple
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); -
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 -
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
Formula
G.f.: RootOf(-_Z+_Z^2+_Z^3+x)-RootOf(-_Z+_Z^2+_Z^3+x)^2-x
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)}
From Seiichi Manyama, Nov 22 2024: (Start)
G.f.: (x*B(x))^3 where B(x) is the g.f. of A001002.
a(n) = 3 * Sum_{k=0..n-3} binomial(n+k,k) * binomial(k,n-3-k)/(n+k). (End)
a(n) ~ 3^(3*n - 5/2) / (sqrt(Pi) * 2^(3/2) * n^(3/2) * 5^(n - 1/2)). - Vaclav Kotesovec, Nov 22 2024
Extensions
More terms from Seiichi Manyama, Nov 21 2024