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 A015033 q-Catalan numbers (binomial version) for q=3. 1

%I

%S 1,1,10,847,627382,4138659802,244829520301060,130191700295480695111,

%T 622829375926755523108996006,26812578369717035183629988539429726,

%U 10387976772168532331015929118843873280496300

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

%H G. C. Greubel, <a href="/A015033/b015033.txt">Table of n, a(n) for n = 0..46</a>

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

%F 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

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

%o (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

%o (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

%Y Cf. A015030 (q=2).

%K nonn,easy

%O 0,3