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A015083 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=2.

%I A015083 #85 Nov 17 2019 13:30:25
%S A015083 1,1,3,17,171,3113,106419,7035649,915028347,236101213721,
%T A015083 121358941877763,124515003203007345,255256125633703622475,
%U A015083 1046039978882750301409545,8571252355254982356001107795,140448544236464264647066322058465,4602498820363674769217316088142020635
%N A015083 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=2.
%C A015083 Limit_{n->inf} a(n)/2^((n-1)(n-2)/2) = Product{k>=1} 1/(1-1/2^k) = 3.462746619455... (cf. A065446). - _Paul D. Hanna_, Jan 24 2005
%C A015083 It appears that the Hankel transform is 2^A002412(n). - _Paul Barry_, Aug 01 2008
%C A015083 Hankel transform of aerated sequence is A125791. - _Paul Barry_, Dec 15 2010
%H A015083 Seiichi Manyama, <a href="/A015083/b015083.txt">Table of n, a(n) for n = 0..81</a>
%H A015083 J. Fürlinger, J. Hofbauer, <a href="http://dx.doi.org/10.1016/0097-3165(85)90089-5">q-Catalan numbers</a>, Journal of Combinatorial Theory, Series A, Volume 40, Issue 2, November 1985, Pages 248-264.
%H A015083 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 A015083 a(n+1) = Sum_{i=0..n} q^i*a(i)*a(n-i) with q=2 and a(0)=1.
%F A015083 G.f. satisfies: A(x) = 1 / (1 - x*A(2*x)) = 1/(1-x/(1-2*x/(1-2^2*x/(1-2^3*x/(1-...))))) (continued fraction). - _Paul D. Hanna_, Jan 24 2005
%F A015083 G.f. satisfies: A(x) = Sum_{n>=0} Product_{k=0..n-1} 2^k*x*A(2^k*x). - _Paul D. Hanna_, May 17 2010
%F A015083 a(n) = the upper left term in M^(n-1), M = the infinite square production matrix:
%F A015083   1,  2,  0,  0,  0, ...
%F A015083   1,  2,  4,  0,  0, ...
%F A015083   1,  2,  4,  8,  0, ...
%F A015083   1,  2,  4,  8, 16, ...
%F A015083   ...
%F A015083 Also, a(n+1) = sum of top row terms of M^(n-1). Example: top row of M^3 = (17, 34, 56, 64, 0, 0, 0, ...); where a(4) = 17 and a(5) = 171 = (17 + 34 + 56 + 64). - _Gary W. Adamson_, Jul 14 2011
%F A015083 G.f.: T(0), where T(k) = 1 - x*(2^k)/(x*(2^k) - 1/T(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Oct 17 2013
%e A015083 G.f. = 1 + x + 3*x^2 + 17*x^3 + 171*x^4 + 3113*x^5 + 106419*x^6 + 7035649*x^7 + ...
%e A015083 From _Seiichi Manyama_, Dec 05 2016: (Start)
%e A015083 a(1) = 1,
%e A015083 a(2) = 2^1 + 1 = 3,
%e A015083 a(3) = 2^3 + 2^2 + 2*2^1 + 1 = 17,
%e A015083 a(4) = 2^6 + 2^5 + 2*2^4 + 3*2^3 + 3*2^2 + 3*2^1 + 1 = 171. (End)
%t A015083 a[n_] := a[n] = Sum[2^i*a[i]*a[n - i - 1], {i, 0, n - 1}];
%t A015083 a[0] = 1; Array[a, 16, 0] (* _Robert G. Wilson v_, Dec 24 2016 *)
%t A015083 m = 17; ContinuedFractionK[If[i == 1, 1, -2^(i-2) x], 1, {i, 1, m}] + O[x]^m // CoefficientList[#, x]& (* _Jean-François Alcover_, Nov 17 2019 *)
%o A015083 (PARI) a(n)=if(n==0,1,sum(i=0,n-1,2^i*a(i)*a(n-1-i))) \\  _Paul D. Hanna_
%o A015083 (PARI) {a(n) = my(A); if( n<1, n==0, A = vector(n, i, 1); for(k=0, n-1, A[k+1] = if( k<1, 1, A[k]*(1+2^k) + sum(i=1, k-1, 2^i * A[i] * A[k-i]))); A[n])}; /* _Michael Somos_, Jan 30 2005 */
%o A015083 (PARI) {a(n) = my(A); if( n<0, 0, A = O(x); for(k=1, n, A = 1 / (1 - x * subst(A, x, 2*x))); polcoeff(A, n))}; /* _Michael Somos_, Jan 30 2005 */
%o A015083 (Ruby)
%o A015083 def A(q, n)
%o A015083   ary = [1]
%o A015083   (1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + q ** j * ary[j] * ary[i - 1 - j]}}
%o A015083   ary
%o A015083 end
%o A015083 def A015083(n)
%o A015083   A(2, n)
%o A015083 end # _Seiichi Manyama_, Dec 24 2016
%Y A015083 Cf. A065446, A227543.
%Y A015083 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), this sequence (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 A015083 Column k=2 of A090182, A290759.
%K A015083 nonn
%O A015083 0,3
%A A015083 _Olivier Gérard_
%E A015083 Offset changed to 0 by _Seiichi Manyama_, Dec 05 2016