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A015099 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=-4.

%I A015099 #34 Oct 24 2018 08:27:05
%S A015099 1,1,-3,-55,3429,885137,-904638963,-3707218743911,60731665539301365,
%T A015099 3980231929565571675617,-1043385959026442521712292579,
%U A015099 -1094071562179856506263860787078039
%N A015099 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=-4.
%H A015099 Seiichi Manyama, <a href="/A015099/b015099.txt">Table of n, a(n) for n = 0..58</a>
%H A015099 Robin Sulzgruber, <a href="http://othes.univie.ac.at/30616/">The Symmetry of the q,t-Catalan Numbers</a>
%F A015099 a(n+1) = Sum_{i=0..n} q^i*a(i)*a(n-i) with q=-4 and a(0)=1.
%F A015099 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 27 2016
%e A015099 G.f. = 1 + x - 3*x^2 - 55*x^3 + 3429*x^4 + 885137*x^5 + ...
%t A015099 a[1] := 1; a[n_] := a[n] = Sum[(-4)^(i - 1)*a[i]*a[n - i], {i, 1, n - 1}]; Array[a, 20, 1] (* _G. C. Greubel_, Dec 24 2016 *)
%o A015099 (Ruby)
%o A015099 def A(q, n)
%o A015099   ary = [1]
%o A015099   (1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + q ** j * ary[j] * ary[i - 1 - j]}}
%o A015099   ary
%o A015099 end
%o A015099 def A015099(n)
%o A015099   A(-4, n)
%o A015099 end # _Seiichi Manyama_, Dec 24 2016
%Y A015099 Cf. A227543.
%Y A015099 Cf. A015108 (q=-11), A015107 (q=-10), A015106 (q=-9), A015105 (q=-8), A015103 (q=-7), A015102 (q=-6), A015100 (q=-5), this sequence (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), A015093 (q=9), A015095 (q=10), A015096 (q=11).
%Y A015099 Column k=4 of A290789.
%K A015099 sign
%O A015099 0,3
%A A015099 _Olivier Gérard_
%E A015099 Offset changed to 0 by _Seiichi Manyama_, Dec 24 2016