OFFSET
0,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..57
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
MATHEMATICA
Table[QBinomial[2n, n, 2]/(2^(n+1) - 1), {n, 0, 20}] (* Vladimir Reshetnikov, Sep 16 2016 *)
PROG
(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
(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
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved