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

0,5

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Formula

Conjecture: a(n) ~ sqrt(3) * 2^(2*n) / (Pi * n^3). - Vaclav Kotesovec, Jan 06 2023

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

Prog

(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)])

Crossrefs

Cf. A000108, A129175 (coefficients of q_Catalan polynomials), A275213.

Sequence in context: A032010 A032028 A190277 * A127384 A337721 A369328

Adjacent sequences: A274879 A274880 A274881 * A274883 A274884 A274885

Keyword

nonn

Author

Peter Luschny, Jul 19 2016

Status

approved