A274882 a(n) is the largest coefficient of q-binomial(2*n, n) / q-binomial(n+1, 1), which are the q-Catalan polynomials.
1, 1, 1, 1, 2, 4, 9, 23, 62, 176, 512, 1551, 4822, 15266, 49141, 160728, 532890, 1785162, 6039328, 20617808, 70951548, 245911020, 857888714, 3010811846, 10624583264, 37680980256, 134260382400, 480440869030, 1726092837412, 6224442777366, 22523780202156
Offset: 0
Keywords
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..200
Programs
-
Maple
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);
-
Mathematica
p[n_] := QBinomial[2n,n,q]/QBinomial[n+1,1,q]; Table[Max[CoefficientList[p[n] // FunctionExpand, q]], {n,0,20}] // Flatten -
Sage
from sage.combinat.q_analogues import q_catalan_number def T(n): return q_catalan_number(n) print([max(T(n)) for n in (0..10)])
Formula
Conjecture: a(n) ~ sqrt(3) * 2^(2*n) / (Pi * n^3). - Vaclav Kotesovec, Jan 06 2023