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

%I A290777 #24 May 09 2020 02:41:07
%S A290777 1,1,3,43,5885,12833546,583552122727,667480099386451779,
%T A290777 22507185898866512901924729,25700910736350654917922270058287454,
%U A290777 1123582754598967452437582737448130799606015691,2098715344599001562385695830901626594365732485934286582686
%N A290777 a(n) = n-th Carlitz-Riordan q-Catalan number (recurrence version) for q = n.
%H A290777 Alois P. Heinz, <a href="/A290777/b290777.txt">Table of n, a(n) for n = 0..36</a>
%H A290777 J. Fürlinger, 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 A290777 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 A290777 a(n) = [x^n] 1/(1-x/(1-n*x/(1-n^2*x/(1-n^3*x/(1-n^4*x/(1- ... )))))).
%F A290777 a(n) = A290759(n,n) = A090182(2n,n).
%F A290777 a(n) ~ n^(n*(n-1)/2). - _Vaclav Kotesovec_, Aug 19 2017
%p A290777 b:= proc(n, k) option remember; `if`(n=0, 1, add(
%p A290777       b(j, k)*b(n-j-1, k)*k^j, j=0..n-1))
%p A290777     end:
%p A290777 a:= n-> b(n$2):
%p A290777 seq(a(n), n=0..12);
%t A290777 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 A290777 (Python)
%o A290777 from sympy.core.cache import cacheit
%o A290777 @cacheit
%o A290777 def b(n, k):
%o A290777     if n == 0:
%o A290777         return 1
%o A290777     return sum(b(j, k) * b(n - j - 1, k) * k**j for j in range(n))
%o A290777 def a(n): return b(n, n)
%o A290777 print([a(n) for n in range(16)]) # _Indranil Ghosh_, Aug 10 2017
%Y A290777 Main diagonal of A290759.
%Y A290777 Cf. A090182, A290786.
%K A290777 nonn
%O A290777 0,3
%A A290777 _Alois P. Heinz_, Aug 10 2017