A015033 -id:A015033 - cp's OEIS frontend

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

Olivier Gérard

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

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.

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

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

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

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

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

A384437 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) is the n-th q-Catalan number for q=k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 5, 1, 1, 1, 10, 93, 14, 1, 1, 1, 17, 847, 6477, 42, 1, 1, 1, 26, 4433, 627382, 1733677, 132, 1, 1, 1, 37, 16401, 18245201, 4138659802, 1816333805, 429, 1, 1, 1, 50, 48205, 256754526, 1197172898385, 244829520301060, 7526310334829, 1430, 1

Offset: 0

Seiichi Manyama, May 29 2025

			Square array begins:
  1,  1,       1,          1,             1,               1, ...
  1,  1,       1,          1,             1,               1, ...
  1,  2,       5,         10,            17,              26, ...
  1,  5,      93,        847,          4433,           16401, ...
  1, 14,    6477,     627382,      18245201,       256754526, ...
  1, 42, 1733677, 4138659802, 1197172898385, 100333200992026, ...

Columns k=0..12 give A000012, A000108, A015030, A015033, A015034, A015035, A015037, A015038, A015039, A015040, A015041, A015042, A015055.
Main diagonal gives A384282.
Cf. A129175

a(n, k) = if(k==1, binomial(2*n, n)/(n+1), (1-k)/(1-k^(n+1))*prod(j=0, n-1, (1-k^(2*n-j))/(1-k^(j+1))));

from sage.combinat.q_analogues import q_catalan_number
def a(n, k): return q_catalan_number(n, k)

A(n,k) = q_binomial(2*n, n, k)/q_binomial(n+1, 1, k).

cp's OEIS Frontend

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

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