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

%I A290786 #28 Apr 23 2021 01:29:51
%S A290786 1,1,-1,-23,3429,8425506,-412878084725,-497641562809372379,
%T A290786 17436260499054618815283977,20503694883570579788445502041773422,
%U A290786 -917439693541287252616828116888122637934368489,-1746281566732870051764961051797990328294109372786185933382
%N A290786 a(n) = n-th Carlitz-Riordan q-Catalan number (recurrence version) for q = -n.
%H A290786 Alois P. Heinz, <a href="/A290786/b290786.txt">Table of n, a(n) for n = 0..36</a>
%H A290786 J. Fürlinger and J. Hofbauer, <a href="http://dx.doi.org/10.1016/0097-3165(85)90089-5">q-Catalan numbers</a>, Journal of Combinatorial Theory, Series A, Volume 40, Issue 2, November 1985, Pages 248-264.
%H A290786 Robin Sulzgruber, <a href="http://othes.univie.ac.at/30616/">The Symmetry of the q,t-Catalan Numbers</a>, Masterarbeit, University of Vienna. Fakultät für Mathematik, 2013.
%F A290786 a(n) = [x^n] 1/(1-x/(1+n*x/(1-n^2*x/(1+n^3*x/(1-n^4*x/(1+ ... )))))).
%F A290786 a(n) = A290789(n,n).
%p A290786 b:= proc(n, k) option remember; `if`(n=0, 1, add(
%p A290786       b(j, k)*b(n-j-1, k)*(-k)^j, j=0..n-1))
%p A290786     end:
%p A290786 a:= n-> b(n$2):
%p A290786 seq(a(n), n=0..12);
%t A290786 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 *)
%o A290786 (Python)
%o A290786 from sympy.core.cache import cacheit
%o A290786 @cacheit
%o A290786 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)])
%o A290786 def a(n): return b(n, n)
%o A290786 print([a(n) for n in range(16)]) # _Indranil Ghosh_, Aug 10 2017
%Y A290786 Main diagonal of A290789.
%Y A290786 Cf. A290777.
%K A290786 sign
%O A290786 0,4
%A A290786 _Alois P. Heinz_, Aug 10 2017