A119365 Generalized Catalan numbers for triangle A119335.

1, 0, 0, 1, 6, 20, 51, 126, 392, 1513, 5877, 21054, 71270, 242463, 863590, 3193737, 11889414, 43783908, 159998493, 586908936, 2175907284, 8138471667, 30541703733, 114620380032, 430344635913, 1619584557885, 6116422089050

Offset: 0

Author

Paul Barry, May 16 2006

Keywords

• easy
• nonn

Comments

Counts rooted planar n-trees whose number of leaves is divisible by 3.

Crossrefs

Cf. A000108, A001263, A119335, A119366, A119367.

Programs

Maple

A119365 := proc(n)
    local k;
    if n = 0 then
        return 1
    end if;
    a := 0 ;
    for k from 0 to n do
        if modp(n-k,3) = 0 then
            a := a+binomial(n,k)*binomial(n,k+1) ;
        end if;
    end do:
    a/n;
end proc:
seq(A119365(n),n=0..40) ; # R. J. Mathar, Oct 30 2014

Mathematica

A119335[n_, k_] := Sum[Binomial[k, 3j] Binomial[n-k, 3j], {j, 0, n-k}];
a[n_] := A119335[2n, n] - A119335[2n, n+1];
Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Sep 14 2023 *)

Formula

a(n) = A119335(2n,n) - A119335(2n,n+1).

a(n) = Sum_{k=0..n} if(mod(n-k,3)=0, (1/n)*C(n,k)*C(n,k+1), 0).

a(n) + A119366(n) + A119367(n) = A000108(n).