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A015092 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=8.

%I A015092 #34 Nov 17 2019 09:29:25
%S A015092 1,1,9,593,304857,1249312673,40939981188777,10732252327798007281,
%T A015092 22507185898866512901924729,377607964391970470904956530918721,
%U A015092 50681683810611444451901001718927186370889
%N A015092 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=8.
%H A015092 Seiichi Manyama, <a href="/A015092/b015092.txt">Table of n, a(n) for n = 0..47</a>
%H A015092 Robin Sulzgruber, <a href="http://othes.univie.ac.at/30616/">The Symmetry of the q,t-Catalan Numbers</a>
%F A015092 a(n+1) = Sum_{i=0..n} q^i*a(i)*a(n-i) with q=8 and a(0)=1.
%F A015092 G.f. satisfies: A(x) = 1 / (1 - x*A(8*x)) = 1/(1-x/(1-8*x/(1-8^2*x/(1-8^3*x/(1-...))))) (continued fraction). - _Seiichi Manyama_, Dec 26 2016
%e A015092 G.f. = 1 + x + 9*x^2 + 593*x^3 + 304857*x^4 + 1249312673*x^5 + ...
%t A015092 a[n_] := a[n] = Sum[8^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 A015092 m = 11; ContinuedFractionK[If[i == 1, 1, -8^(i - 2) x], 1, {i, 1, m}] + O[x]^m // CoefficientList[#, x]& (* _Jean-François Alcover_, Nov 17 2019 *)
%o A015092 (Ruby)
%o A015092 def A(q, n)
%o A015092   ary = [1]
%o A015092   (1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + q ** j * ary[j] * ary[i - 1 - j]}}
%o A015092   ary
%o A015092 end
%o A015092 def A015092(n)
%o A015092   A(8, n)
%o A015092 end # _Seiichi Manyama_, Dec 24 2016
%Y A015092 Cf. A227543.
%Y A015092 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), this sequence (q=8), A015093 (q=9), A015095 (q=10), A015096 (q=11).
%Y A015092 Column k=8 of A090182, A290759.
%K A015092 nonn
%O A015092 0,3
%A A015092 _Olivier Gérard_
%E A015092 Offset changed to 0 by _Seiichi Manyama_, Dec 24 2016