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A015084 Carlitz-Riordan q-Catalan numbers for q=3.

%I A015084 #79 Nov 17 2019 09:28:49
%S A015084 1,1,4,43,1252,104098,25511272,18649337311,40823535032644,
%T A015084 267924955577741566,5274102955963545775864,
%U A015084 311441054994969341088610030,55171471477692117486494217498280
%N A015084 Carlitz-Riordan q-Catalan numbers for q=3.
%C A015084 Limit_{n->inf} a(n)/3^((n-1)(n-2)/2) = Product{k>=1} 1/(1-1/3^k) = 1.785312341998534190367486296013703535718796... - _Paul D. Hanna_, Jan 24 2005
%C A015084 It appears that the Hankel transform is 3^A002412(n). - _Paul Barry_, Aug 01 2008
%C A015084 Hankel transform of the aerated sequence is 3^C(n+1,3). - _Paul Barry_, Oct 31 2008
%H A015084 Seiichi Manyama, <a href="/A015084/b015084.txt">Table of n, a(n) for n = 0..65</a>
%H A015084 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 A015084 a(n+1) = Sum_{i=0..n} q^i*a(i)*a(n-i) with q=3 and a(0)=1.
%F A015084 G.f. satisfies: A(x) = 1/(1-x*A(3*x)) = 1/(1-x/(1-3*x/(1-3^2*x/(1-3^3*x/(1-...))))) (continued fraction). - _Paul D. Hanna_, Jan 24 2005
%F A015084 a(n) = the upper left term in M^n, M an infinite production matrix as follows:
%F A015084   1,  3,  0,  0,  0,  0, ...
%F A015084   1,  3,  9,  0,  0,  0, ...
%F A015084   1,  3,  9, 27,  0,  0, ...
%F A015084   1,  3,  9, 27, 81,  0, ...
%F A015084   ... - _Gary W. Adamson_, Jul 14 2011
%F A015084 G.f.: T(0), where T(k) = 1 - x*3^k/(x*3^k - 1/T(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Oct 18 2013
%e A015084 G.f. = 1 + x + 4*x^2 + 43*x^3 + 1252*x^4 + 104098*x^5 + 25511272*x^6 + ...
%e A015084 From _Seiichi Manyama_, Dec 05 2016: (Start)
%e A015084 a(1) = 1,
%e A015084 a(2) = 3^1 + 1 = 4,
%e A015084 a(3) = 3^3 + 3^2 + 2*3^1 + 1 = 43,
%e A015084 a(4) = 3^6 + 3^5 + 2*3^4 + 3*3^3 + 3*3^2 + 3*3^1 + 1 = 1252. (End)
%p A015084 A015084 := proc(n)
%p A015084     option remember;
%p A015084     if n = 1 then
%p A015084         1;
%p A015084     else
%p A015084     add(3^(i-1)*procname(i)*procname(n-i),i=1..n-1) ;
%p A015084     end if;
%p A015084 end proc: # _R. J. Mathar_, Sep 29 2012
%t A015084 a[n_] := a[n] = Sum[3^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 A015084 m = 13; ContinuedFractionK[If[i == 1, 1, -3^(i-2) x], 1, {i, 1, m}] + O[x]^m // CoefficientList[#, x]& (* _Jean-François Alcover_, Nov 17 2019 *)
%o A015084 (PARI) a(n)=if(n==1,1,sum(i=1,n-1,3^(i-1)*a(i)*a(n-i))) \\ _Paul D. Hanna_
%o A015084 (Ruby)
%o A015084 def A(q, n)
%o A015084   ary = [1]
%o A015084   (1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + q ** j * ary[j] * ary[i - 1 - j]}}
%o A015084   ary
%o A015084 end
%o A015084 def A015084(n)
%o A015084   A(3, n)
%o A015084 end # _Seiichi Manyama_, Dec 24 2016
%Y A015084 Cf. A227543.
%Y A015084 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), this sequence (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 A015084 Column k=3 of A090182, A290759.
%K A015084 nonn
%O A015084 0,3
%A A015084 _Olivier Gérard_
%E A015084 More terms from _Paul D. Hanna_, Jan 24 2005
%E A015084 Offset changed to 0 by _Seiichi Manyama_, Dec 05 2016