A015085 - OEIS

%I #62 Nov 03 2021 07:56:58

%S 1,1,5,89,5885,1518897,1558435125,6386478643785,104648850228298925,

%T 6858476391221411106209,1797922152786660462507074405,

%U 1885261615172756172119161342909753

%N Carlitz-Riordan q-Catalan numbers (recurrence version) for q=4.

%H Seiichi Manyama, <a href="/A015085/b015085.txt">Table of n, a(n) for n = 0..58</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=4 and a(0)=1.

%F G.f. satisfies: A(x) = 1 / (1 - x*A(4*x)) = 1/(1-x/(1-4*x/(1-4^2*x/(1-4^3*x/(1-...))))) (continued fraction). - _Seiichi Manyama_, Dec 26 2016

%F a(n) ~ c * 2^(n*(n-1)), where c = Product{j>=1} 1/(1-1/4^j) = 1/QPochhammer(1/4) = 1.4523536424495970158347130224852748733612279788... - _Vaclav Kotesovec_, Nov 03 2021

%e G.f. = 1 + x + 5*x^2 + 89*x^3 + 5885*x^4 + 1518897*x^5 + 1558435125*x^6 + ...

%e From _Seiichi Manyama_, Dec 05 2016: (Start)

%e a(1) = 1,

%e a(2) = 4^1 + 1 = 5,

%e a(3) = 4^3 + 4^2 + 2*4^1 + 1 = 89,

%e a(4) = 4^6 + 4^5 + 2*4^4 + 3*4^3 + 3*4^2 + 3*4^1 + 1 = 5885. (End)

%t a[n_] := a[n] = Sum[4^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 = 12; ContinuedFractionK[If[i == 1, 1, -4^(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 A015085(n)

%o   A(4, 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), this sequence (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=4 of A090182, A290759.

%K nonn

%O 0,3

%A _Olivier Gérard_

%E Offset changed to 0 by _Seiichi Manyama_, Dec 05 2016