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%I #30 Nov 17 2019 09:30:32
%S 1,1,-8,-665,483544,3173511682,-187386353065808,-99585165693268026701,
%T 476312561203989614441440600,20503694883570579788445502041773422,
%U -7943551457092331370323478258038812629918704
%N Carlitz-Riordan q-Catalan numbers (recurrence version) for q=-9.
%H Seiichi Manyama, <a href="/A015106/b015106.txt">Table of n, a(n) for n = 0..46</a>
%H 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 a(n+1) = Sum_{i=0..n} q^i*a(i)*a(n-i) with q=-9 and a(0)=1.
%F G.f. satisfies: A(x) = 1 / (1 - x*A(-9*x)) = 1/(1-x/(1+9*x/(1-9^2*x/(1+9^3*x/(1-...))))) (continued fraction). - _Seiichi Manyama_, Dec 28 2016
%e G.f. = 1 + x - 8*x^2 - 665*x^3 + 483544*x^4 + 3173511682*x^5 + ...
%t m = 11; ContinuedFractionK[If[i == 1, 1, -(-9)^(i - 2) x], 1, {i, 1, m}] + O[x]^m // CoefficientList[#, x]& (* _Jean-François Alcover_, Nov 17 2019 *)
%o (Ruby)
%o def A(q, n)
%o ary = [1]
%o (1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + q ** j * ary[j] * ary[i - 1 - j]}}
%o ary
%o end
%o def A015106(n)
%o A(-9, n)
%o end # _Seiichi Manyama_, Dec 25 2016
%Y Cf. A227543.
%Y Cf. A015108 (q=-11), A015107 (q=-10), this sequence (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), A015092 (q=8), A015093 (q=9), A015095 (q=10), A015096 (q=11).
%Y Column k=9 of A290789.
%K sign
%O 0,3
%A _Olivier Gérard_
%E Offset changed to 0 by _Seiichi Manyama_, Dec 25 2016
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