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A015093 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=9.

%I A015093 #41 Nov 17 2019 09:29:33
%S A015093 1,1,10,829,606070,3977651242,234884294434900,124827614155955343925,
%T A015093 597046858511123656669455550,25700910736350654917922270058287454,
%U A015093 9957059456624152426469878400757673046606860
%N A015093 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=9.
%H A015093 Seiichi Manyama, <a href="/A015093/b015093.txt">Table of n, a(n) for n = 0..46</a>
%H A015093 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 A015093 a(n+1) = Sum_{i=0..n} q^i*a(i)*a(n-i) with q=9 and a(0)=1.
%F A015093 G.f. satisfies: A(x) = 1 / (1 - x*A(9*x)) = 1/(1-x/(1-9*x/(1-9^2*x/(1-9^3*x/(1-...))))) (continued fraction). - _Seiichi Manyama_, Dec 27 2016
%e A015093 G.f. = 1 + x + 10*x^2 + 829*x^3 + 606070*x^4 + 3977651242*x^5 + ...
%t A015093 a[n_] := a[n] = Sum[9^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 A015093 m = 11; ContinuedFractionK[If[i == 1, 1, -9^(i - 2) x], 1, {i, 1, m}] + O[x]^m // CoefficientList[#, x]& (* _Jean-François Alcover_, Nov 17 2019 *)
%o A015093 (Ruby)
%o A015093 def A(q, n)
%o A015093   ary = [1]
%o A015093   (1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + q ** j * ary[j] * ary[i - 1 - j]}}
%o A015093   ary
%o A015093 end
%o A015093 def A015093(n)
%o A015093   A(9, n)
%o A015093 end # _Seiichi Manyama_, Dec 24 2016
%Y A015093 Cf. A227543.
%Y A015093 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), A015089 (q=6), A015091 (q=7), A015092 (q=8), this sequence (q=9), A015095 (q=10), A015096 (q=11).
%Y A015093 Column k=9 of A090182, A290759.
%K A015093 nonn
%O A015093 0,3
%A A015093 _Olivier Gérard_
%E A015093 Offset changed to 0 by _Seiichi Manyama_, Dec 24 2016