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A015089 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=6.

%I A015089 #37 Nov 17 2019 09:29:11
%S A015089 1,1,7,265,57799,75025897,583552122727,27227375795690569,
%T A015089 7621977131953256556295,12802009986716861649949951657,
%U A015089 129014790439200398432389878440405671
%N A015089 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=6.
%H A015089 Seiichi Manyama, <a href="/A015089/b015089.txt">Table of n, a(n) for n = 0..51</a>
%H A015089 Robin Sulzgruber, <a href="https://doi.org/10.25365/thesis.30616">The Symmetry of the q,t-Catalan Numbers</a>, Thesis, University of Vienna, 2013.
%F A015089 a(n+1) = Sum_{i=0..n} q^i*a(i)*a(n-i) with q=6 and a(0)=1.
%F A015089 G.f. satisfies: A(x) = 1 / (1 - x*A(6*x)) = 1/(1-x/(1-6*x/(1-6^2*x/(1-6^3*x/(1-...))))) (continued fraction). - _Seiichi Manyama_, Dec 26 2016
%e A015089 G.f. = 1 + x + 7*x^2 + 265*x^3 + 57799*x^4 + 75025897*x^5 + 583552122727*x^6 + ...
%t A015089 a[n_] := a[n] = Sum[6^i*a[i]*a[n -i -1], {i, 0, n -1}]; a[0] = 1; Array[a, 16, 0] (* _Robert G. Wilson v_, Dec 24 2016 *)
%t A015089 m = 11; ContinuedFractionK[If[i == 1, 1, -6^(i - 2) x], 1, {i, 1, m}] + O[x]^m // CoefficientList[#, x]& (* _Jean-François Alcover_, Nov 17 2019 *)
%o A015089 (Ruby)
%o A015089 def A(q, n)
%o A015089   ary = [1]
%o A015089   (1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + q ** j * ary[j] * ary[i - 1 - j]}}
%o A015089   ary
%o A015089 end
%o A015089 def A015089(n)
%o A015089   A(6, n)
%o A015089 end # _Seiichi Manyama_, Dec 24 2016
%Y A015089 Cf. A227543.
%Y A015089 Cf. A015108 (q=-11), A015107 (q=-10), A015106 (q=-9), A015105 (q=-8), A015103 (q=-7), A015102 (q=-6), A015100 (q=-5), A015099 (q=-4), A015098 (q=-3), A015097 (q=-2), A090192 (q=-1), A000108 (q=1), A015083 (q=2), A015084 (q=3), A015085 (q=4), A015086 (q=5), this sequence (q=6), A015091 (q=7), A015092 (q=8), A015093 (q=9), A015095 (q=10), A015096 (q=11).
%Y A015089 Column k=6 of A090182, A290759.
%K A015089 nonn
%O A015089 0,3
%A A015089 _Olivier Gérard_
%E A015089 Offset changed to 0 by _Seiichi Manyama_, Dec 24 2016