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

0,5

Comments

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

Links

Table of n, a(n) for n=0..26.

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).

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 *)

Crossrefs

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

Sequence in context: A267168 A266760 A213586 * A001211 A122225 A275112

Adjacent sequences: A119362 A119363 A119364 * A119366 A119367 A119368

Keyword

easy,nonn

Author

Paul Barry, May 16 2006

Status

approved