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%I #23 Jan 06 2023 16:22:38
%S 1,1,1,1,2,4,9,23,62,176,512,1551,4822,15266,49141,160728,532890,
%T 1785162,6039328,20617808,70951548,245911020,857888714,3010811846,
%U 10624583264,37680980256,134260382400,480440869030,1726092837412,6224442777366,22523780202156
%N a(n) is the largest coefficient of q-binomial(2*n, n) / q-binomial(n+1, 1), which are the q-Catalan polynomials.
%H Vaclav Kotesovec, <a href="/A274882/b274882.txt">Table of n, a(n) for n = 0..200</a>
%F Conjecture: a(n) ~ sqrt(3) * 2^(2*n) / (Pi * n^3). - _Vaclav Kotesovec_, Jan 06 2023
%p with(QDifferenceEquations): MaxQCatalan := proc(n) local P; P := f -> expand(simplify(expand(f))); P(QBinomial(2*n,n,q)/QBrackets(n+1,q)); max(seq(coeff(%,q,j), j=0..degree(%))) end: seq(MaxQCatalan(n), n=0..20);
%t p[n_] := QBinomial[2n,n,q]/QBinomial[n+1,1,q]; Table[Max[CoefficientList[p[n] // FunctionExpand, q]], {n,0,20}] // Flatten
%o (Sage)
%o from sage.combinat.q_analogues import q_catalan_number
%o def T(n): return q_catalan_number(n)
%o print([max(T(n)) for n in (0..10)])
%Y Cf. A000108, A129175 (coefficients of q_Catalan polynomials), A275213.
%K nonn
%O 0,5
%A _Peter Luschny_, Jul 19 2016
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