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A015085 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=4.

%I A015085 #62 Nov 03 2021 07:56:58
%S A015085 1,1,5,89,5885,1518897,1558435125,6386478643785,104648850228298925,
%T A015085 6858476391221411106209,1797922152786660462507074405,
%U A015085 1885261615172756172119161342909753
%N A015085 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=4.
%H A015085 Seiichi Manyama, <a href="/A015085/b015085.txt">Table of n, a(n) for n = 0..58</a>
%H A015085 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 A015085 a(n+1) = Sum_{i=0..n} q^i*a(i)*a(n-i) with q=4 and a(0)=1.
%F A015085 G.f. satisfies: A(x) = 1 / (1 - x*A(4*x)) = 1/(1-x/(1-4*x/(1-4^2*x/(1-4^3*x/(1-...))))) (continued fraction). - _Seiichi Manyama_, Dec 26 2016
%F A015085 a(n) ~ c * 2^(n*(n-1)), where c = Product{j>=1} 1/(1-1/4^j) = 1/QPochhammer(1/4) = 1.4523536424495970158347130224852748733612279788... - _Vaclav Kotesovec_, Nov 03 2021
%e A015085 G.f. = 1 + x + 5*x^2 + 89*x^3 + 5885*x^4 + 1518897*x^5 + 1558435125*x^6 + ...
%e A015085 From _Seiichi Manyama_, Dec 05 2016: (Start)
%e A015085 a(1) = 1,
%e A015085 a(2) = 4^1 + 1 = 5,
%e A015085 a(3) = 4^3 + 4^2 + 2*4^1 + 1 = 89,
%e A015085 a(4) = 4^6 + 4^5 + 2*4^4 + 3*4^3 + 3*4^2 + 3*4^1 + 1 = 5885. (End)
%t A015085 a[n_] := a[n] = Sum[4^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 A015085 m = 12; ContinuedFractionK[If[i == 1, 1, -4^(i - 2) x], 1, {i, 1, m}] + O[x]^m // CoefficientList[#, x]& (* _Jean-François Alcover_, Nov 17 2019 *)
%o A015085 (Ruby)
%o A015085 def A(q, n)
%o A015085   ary = [1]
%o A015085   (1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + q ** j * ary[j] * ary[i - 1 - j]}}
%o A015085   ary
%o A015085 end
%o A015085 def A015085(n)
%o A015085   A(4, n)
%o A015085 end # _Seiichi Manyama_, Dec 24 2016
%Y A015085 Cf. A227543.
%Y A015085 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), this sequence (q=4), A015086 (q=5), A015089 (q=6), A015091 (q=7), A015092 (q=8), A015093 (q=9), A015095 (q=10), A015096 (q=11).
%Y A015085 Column k=4 of A090182, A290759.
%K A015085 nonn
%O A015085 0,3
%A A015085 _Olivier Gérard_
%E A015085 Offset changed to 0 by _Seiichi Manyama_, Dec 05 2016