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%I #43 Nov 17 2019 09:29:18
%S 1,1,8,407,140456,337520898,5673390747984,667480099386451779,
%T 549699898523248769128232,3168911624115201777713785471406,
%U 127877020635106970108300418456422667248
%N Carlitz-Riordan q-Catalan numbers (recurrence version) for q=7.
%H Seiichi Manyama, <a href="/A015091/b015091.txt">Table of n, a(n) for n = 0..49</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=7 and a(0)=1.
%F G.f. satisfies: A(x) = 1 / (1 - x*A(7*x)) = 1/(1-x/(1-7*x/(1-7^2*x/(1-7^3*x/(1-...))))) (continued fraction). - _Seiichi Manyama_, Dec 26 2016
%e G.f. = 1 + x + 8*x^2 + 407*x^3 + 140456*x^4 + 337520898*x^5 + ...
%t a[n_] := a[n] = Sum[7^i*a[i]*a[n -i -1], {i, 0, n -1}]; a[0] = 1; Array[a, 16, 0] (* _Robert G. Wilson v_, Dec 24 2016 *)
%t m = 11; ContinuedFractionK[If[i == 1, 1, -7^(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 A015091(n)
%o A(7, n)
%o end # _Seiichi Manyama_, Dec 24 2016
%Y Cf. A227543.
%Y 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), A015097 (q=-2), A090192 (q=-1), A000108 (q=1), A015083 (q=2), A015084 (q=3), A015085 (q=4), A015086 (q=5), A015089 (q=6), this sequence (q=7), A015092 (q=8), A015093 (q=9), A015095 (q=10), A015096 (q=11).
%Y Column k=7 of A090182, A290759.
%K nonn
%O 0,3
%A _Olivier Gérard_
%E Offset changed to 0 by _Seiichi Manyama_, Dec 24 2016
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