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A015098 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=-3.

%I A015098 #40 Nov 17 2019 09:30:00
%S A015098 1,1,-2,-23,586,48778,-11759396,-8596478231,18783386191762,
%T A015098 123275424165263086,-2426183754235085042972,
%U A015098 -143268577734839493464012630,25379312219817753259837452498340
%N A015098 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=-3.
%H A015098 Seiichi Manyama, <a href="/A015098/b015098.txt">Table of n, a(n) for n = 0..65</a>
%H A015098 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 A015098 a(n+1) = Sum_{i=0..n} q^i*a(i)*a(n-i) with q=-3 and a(0)=1.
%F A015098 G.f. satisfies: A(x) = 1 / (1 - x*A(-3*x)) = 1/(1-x/(1+3*x/(1-3^2*x/(1+3^3*x/(1-...))))) (continued fraction). - _Seiichi Manyama_, Dec 27 2016
%e A015098 G.f. = 1 + x - 2*x^2 - 23*x^3 + 586*x^4 + 48778*x^5 + ...
%t A015098 a[1] := 1; a[n_] := a[n] = Sum[(-3)^(i - 1)*a[i]*a[n - i], {i, 1, n - 1}]; Array[a, 20, 1] (* _G. C. Greubel_, Dec 24 2016 *)
%t A015098 m = 13; ContinuedFractionK[If[i == 1, 1, (-1)^(i+1) 3^(i-2) x], 1, {i, 1, m}] + O[x]^m // CoefficientList[#, x]& (* _Jean-François Alcover_, Nov 17 2019 *)
%o A015098 (Ruby)
%o A015098 def A(q, n)
%o A015098   ary = [1]
%o A015098   (1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + q ** j * ary[j] * ary[i - 1 - j]}}
%o A015098   ary
%o A015098 end
%o A015098 def A015098(n)
%o A015098   A(-3, n)
%o A015098 end # _Seiichi Manyama_, Dec 24 2016
%Y A015098 Cf. A227543.
%Y A015098 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), this sequence (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), A015093 (q=9), A015095 (q=10), A015096 (q=11).
%Y A015098 Column k=3 of A290789.
%K A015098 sign
%O A015098 0,3
%A A015098 _Olivier Gérard_
%E A015098 Offset changed to 0 by _Seiichi Manyama_, Dec 24 2016