%I #37 Nov 17 2019 09:30:14

%S 1,1,-5,-191,40915,53110057,-412878084725,-19264066381851695,

%T 5392667163887921078275,9057620836725683164283293369,

%U -91279931160615494871228103624209605

%N Carlitz-Riordan q-Catalan numbers (recurrence version) for q=-6.

%H Seiichi Manyama, <a href="/A015102/b015102.txt">Table of n, a(n) for n = 0..51</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=-6 and a(0)=1.

%F G.f. satisfies: A(x) = 1 / (1 - x*A(-6*x)) = 1/(1-x/(1+6*x/(1-6^2*x/(1+6^3*x/(1-...))))) (continued fraction). - _Seiichi Manyama_, Dec 27 2016

%e G.f. = 1 + x - 5*x^2 - 191*x^3 + 40915*x^4 + 53110057*x^5 + ...

%t a[1] := 1; a[n_] := a[n] = Sum[(-6)^(i - 1)*a[i]*a[n - i], {i, 1, n - 1}]; Array[a, 12, 1] (* _G. C. Greubel_, Dec 24 2016 *)

%t m = 11; ContinuedFractionK[If[i == 1, 1, -(-6)^(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 A015102(n)

%o A(-6, 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), this sequence (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=6 of A290789.

%K sign

%O 0,3

%A _Olivier Gérard_

%E Offset changed to 0 by _Seiichi Manyama_, Dec 24 2016