A290786 a(n) = n-th Carlitz-Riordan q-Catalan number (recurrence version) for q = -n.

1, 1, -1, -23, 3429, 8425506, -412878084725, -497641562809372379, 17436260499054618815283977, 20503694883570579788445502041773422, -917439693541287252616828116888122637934368489, -1746281566732870051764961051797990328294109372786185933382

Offset: 0

Author

Alois P. Heinz, Aug 10 2017

Keywords

• sign

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..36
• J. Fürlinger and 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 A290789.

Cf. A290777.

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): return 1 if n==0 else 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) = A290789(n,n).