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A015097 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=-2.

%I A015097 #49 Mar 09 2021 19:11:52
%S A015097 1,1,-1,-7,47,873,-26433,-1749159,220526159,56904690761,
%T A015097 -29022490524961,-29777360924913095,60924625361199230575,
%U A015097 249669263740090899509545,-2044791574538659983034398465,-33505955988983997787211823466215
%N A015097 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=-2.
%H A015097 Seiichi Manyama, <a href="/A015097/b015097.txt">Table of n, a(n) for n = 0..81</a>
%H A015097 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 A015097 a(n+1) = Sum_{i=0..n} q^i*a(i)*a(n-i) with q=-2 and a(0)=1.
%F A015097 G.f: 1/(1-x/(1+2x/(1-4x/(1+8x/(1-16x/(1+... (continued fraction). - _Paul Barry_, Jan 15 2009
%F A015097 G.f. satisfies: A(x) = 1 / (1 - x*A(-2*x)). - _Seiichi Manyama_, Dec 27 2016
%e A015097 G.f. = 1 + x - x^2 - 7*x^3 + 47*x^4 + 873*x^5 + ...
%t A015097 m = 16;
%t A015097 ContinuedFractionK[If[i == 1, 1, (-1)^(i+1) 2^(i-2) x], 1, {i, 1, m}] + O[x]^m // CoefficientList[#, x]& (* _Jean-François Alcover_, Nov 17 2019 *)
%o A015097 (Ruby)
%o A015097 def A(q, n)
%o A015097   ary = [1]
%o A015097   (1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + q ** j * ary[j] * ary[i - 1 - j]}}
%o A015097   ary
%o A015097 end
%o A015097 def A015097(n)
%o A015097   A(-2, n)
%o A015097 end # _Seiichi Manyama_, Dec 24 2016
%o A015097 (Python)
%o A015097 l=[1]
%o A015097 for n in range(1, 21):
%o A015097     l.append(sum([(-2)**i*l[i]*l[n - 1 - i] for i in range(n)]))
%o A015097 print(l) # _Indranil Ghosh_, Aug 14 2017
%Y A015097 Cf. A227543.
%Y A015097 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), this sequence (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 A015097 Column k=2 of A290789.
%K A015097 sign
%O A015097 0,4
%A A015097 _Olivier Gérard_
%E A015097 Offset changed to 0 by _Seiichi Manyama_, Dec 24 2016