A290777 a(n) = n-th Carlitz-Riordan q-Catalan number (recurrence version) for q = n.

1, 1, 3, 43, 5885, 12833546, 583552122727, 667480099386451779, 22507185898866512901924729, 25700910736350654917922270058287454, 1123582754598967452437582737448130799606015691, 2098715344599001562385695830901626594365732485934286582686

Offset: 0

Author

Alois P. Heinz, Aug 10 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..36
• J. Fürlinger, J. Hofbauer, q-Catalan numbers, Journal of Combinatorial Theory, Series A, Volume 40, Issue 2, November 1985, Pages 248-264.
• Robin Sulzgruber, The Symmetry of the q,t-Catalan Numbers, Masterarbeit, University of Vienna. Fakultät für Mathematik, 2013.

Crossrefs

Main diagonal of A290759.

Cf. A090182, A290786.

Programs

Maple

b:= proc(n, k) option remember; `if`(n=0, 1, add(
      b(j, k)*b(n-j-1, k)*k^j, j=0..n-1))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..12);

Mathematica

b[n_, k_]:=b[n, k]=If[n==0, 1, Sum[b[j, k] b[n - j - 1, k]*k^j, {j, 0, n - 1}]]; Table[b[n, n], {n, 0, 15}] (* Indranil Ghosh, Aug 10 2017 *)

Python

from sympy.core.cache import cacheit
@cacheit
def b(n, k):
    if n == 0:
        return 1
    return sum(b(j, k) * b(n - j - 1, k) * k**j for j in range(n))
def a(n): return b(n, n)
print([a(n) for n in range(16)]) # Indranil Ghosh, Aug 10 2017

Formula

a(n) = [x^n] 1/(1-x/(1-n*x/(1-n^2*x/(1-n^3*x/(1-n^4*x/(1- ... )))))).

a(n) = A290759(n,n) = A090182(2n,n).

a(n) ~ n^(n*(n-1)/2). - Vaclav Kotesovec, Aug 19 2017