A015033 q-Catalan numbers (binomial version) for q=3.

1, 1, 10, 847, 627382, 4138659802, 244829520301060, 130191700295480695111, 622829375926755523108996006, 26812578369717035183629988539429726, 10387976772168532331015929118843873280496300

Offset: 0

Author

Olivier Gérard

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..46

Crossrefs

Cf. A015030 (q=2).

Programs

Magma

q:=3; [1] cat [((1-q)/(1-q^(n+1)))*(&*[(1-q^(2*n-k))/(1-q^(k+1)): k in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Nov 11 2018

Mathematica

Table[2 QBinomial[2n, n, 3]/(3^(n+1) - 1), {n, 0, 20}]

PARI

q=3; for(n=0, 20, print1(((1-q)/(1-q^(n+1)))*prod(k=0,n-1, (1-q^(2*n-k))/(1-q^(k+1))), ", ")) \\ G. C. Greubel, Nov 11 2018

Formula

a(n) = binomial(2*n, n, q)/(n+1)_q, where binomial(n,m,q) is the q-binomial coefficient, with q=3.

a(n) = ((1-q)/(1-q^(n+1)))*Product_{k=0..(n-1)} (1-q^(2*n-k))/(1-q^(k+1)), with q=3. - G. C. Greubel, Nov 11 2018