A015034 q-Catalan numbers (binomial version) for q=4.

1, 1, 17, 4433, 18245201, 1197172898385, 1255709588423576145, 21068918017101222558779985, 5655752483351603939678821837720145, 24291387778773301588924456932322615789898321

Offset: 0

Author

Olivier Gérard

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..41

Crossrefs

Cf. A015030 (q=2).

Programs

Magma

q:=4; [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[3*QBinomial[2 n, n, 4]/(4^(n + 1) - 1), {n, 0, 20}] (* G. C. Greubel, Nov 11 2018 *)

PARI

q=4; 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

Formula

a(n) = binomial(2*n, n, q)/(n+1)_q, where binomial(n,m,q) is the q-binomial coefficient, with q=4.

a(n) = ((1-q)/(1-q^(n+1)))*Product_{k=0..(n-1)} (1-q^(2*n-k))/(1-q^(k+1)), with q=4. - G. C. Greubel, Nov 11 2018