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A015100 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=-5.

%I A015100 #39 Nov 17 2019 09:30:07
%S A015100 1,1,-4,-109,13436,8425506,-26312994024,-411193252871529,
%T A015100 32123650782112689116,12548365338592689141400286,
%U A015100 -24508500955561451477156078353144
%N A015100 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=-5.
%H A015100 Seiichi Manyama, <a href="/A015100/b015100.txt">Table of n, a(n) for n = 0..53</a>
%H A015100 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 A015100 a(n+1) = Sum_{i=0..n} q^i*a(i)*a(n-i) with q=-5 and a(0)=1.
%F A015100 G.f. satisfies: A(x) = 1 / (1 - x*A(-5*x)) = 1/(1-x/(1+5*x/(1-5^2*x/(1+5^3*x/(1-...))))) (continued fraction). - _Seiichi Manyama_, Dec 27 2016
%e A015100 G.f. = 1 + x - 5*x^2 - 191*x^3 + 40915*x^4 + 53110057*x^5 + ...
%t A015100 a[1] := 1; a[n_] := a[n] = Sum[(-5)^(i - 1)*a[i]*a[n - i], {i, 1, n - 1}]; Array[a, 20, 1] (* _G. C. Greubel_, Dec 24 2016 *)
%t A015100 m = 11; ContinuedFractionK[If[i == 1, 1, -(-5)^(i - 2) x], 1, {i, 1, m}] + O[x]^m // CoefficientList[#, x]& (* _Jean-François Alcover_, Nov 17 2019 *)
%o A015100 (Ruby)
%o A015100 def A(q, n)
%o A015100   ary = [1]
%o A015100   (1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + q ** j * ary[j] * ary[i - 1 - j]}}
%o A015100   ary
%o A015100 end
%o A015100 def A015100(n)
%o A015100   A(-5, n)
%o A015100 end # _Seiichi Manyama_, Dec 24 2016
%Y A015100 Cf. A227543.
%Y A015100 Cf. A015108 (q=-11), A015107 (q=-10), A015106 (q=-9), A015105 (q=-8), A015103 (q=-7), A015102 (q=-6), this sequence (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 A015100 Column k=5 of A290789.
%K A015100 sign
%O A015100 0,3
%A A015100 _Olivier Gérard_
%E A015100 Offset changed to 0 by _Seiichi Manyama_, Dec 24 2016