A015030 q-Catalan numbers (binomial version) for q=2.

1, 1, 5, 93, 6477, 1733677, 1816333805, 7526310334829, 124031223014725741, 8152285307423733458541, 2140200604371078953284092525, 2245805993494514875022552272042605, 9423041917569791458584837551185555483245, 158121354267437848361217045222877873550507035245

Offset: 0

Author

Olivier Gérard

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..57

Crossrefs

Cf. A065446, A136097.

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

Programs

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

Mathematica

Table[QBinomial[2n, n, 2]/(2^(n+1) - 1), {n, 0, 20}] (* Vladimir Reshetnikov, Sep 16 2016 *)

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

Sage

from sage.combinat.q_analogues import q_catalan_number
[q_catalan_number(n, 2) for n in range(20)] # G. C. Greubel, Nov 21 2018

Formula

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

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

a(n) ~ c * 2^(n^2-n-1), where c = 3.462746619455... = A065446. - Vladimir Reshetnikov, Sep 26 2021

a(n) = (-1)^n * A136097(n). - Michael Somos, Jan 10 2023

a(n) = Product_{1 <= i <= j <= n-1} (2^(i+j+2) - 1)/(2^(i+j) - 1). - Peter Bala, Feb 24 2023