%I #44 Feb 27 2023 18:06:29

%S 1,1,5,93,6477,1733677,1816333805,7526310334829,124031223014725741,

%T 8152285307423733458541,2140200604371078953284092525,

%U 2245805993494514875022552272042605,9423041917569791458584837551185555483245,158121354267437848361217045222877873550507035245

%N q-Catalan numbers (binomial version) for q=2.

%H G. C. Greubel, <a href="/A015030/b015030.txt">Table of n, a(n) for n = 0..57</a>

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

%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=2. - _G. C. Greubel_, Nov 11 2018

%F a(n) ~ c * 2^(n^2-n-1), where c = 3.462746619455... = A065446. - _Vladimir Reshetnikov_, Sep 26 2021

%F a(n) = (-1)^n * A136097(n). - _Michael Somos_, Jan 10 2023

%F a(n) = Product_{1 <= i <= j <= n-1} (2^(i+j+2) - 1)/(2^(i+j) - 1). - _Peter Bala_, Feb 24 2023

%t Table[QBinomial[2n, n, 2]/(2^(n+1) - 1), {n, 0, 20}] (* _Vladimir Reshetnikov_, Sep 16 2016 *)

%o (PARI) q=2; 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:=2; [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

%o (Sage)

%o from sage.combinat.q_analogues import q_catalan_number

%o [q_catalan_number(n, 2) for n in range(20)] # _G. C. Greubel_, Nov 21 2018

%Y Cf. A065446, A136097.

%Y q-Catalan numbers for q = 3..12 and q = -2..-11: A015033 - A015035, A015037 - A015042, A015055 - A015058, A015060 - A015062, A015072, A015077 - A015079.

%K nonn,easy

%O 0,3

%A _Olivier Gérard_