A277218 Maximal coefficient among the polynomials in row n of the triangle of q-binomial coefficients.

1, 1, 1, 1, 2, 2, 3, 5, 8, 12, 20, 32, 58, 94, 169, 289, 526, 910, 1667, 2934, 5448, 9686, 18084, 32540, 61108, 110780, 208960, 381676, 723354, 1328980, 2527074, 4669367, 8908546, 16535154, 31630390, 58965214, 113093022, 211591218, 406680465, 763535450, 1470597342, 2769176514, 5342750699, 10089240974

Offset: 0

Author

Vladimir Reshetnikov, Oct 05 2016

Keywords

• nonn

Comments

q-binomial coefficients are polynomials in q with integer coefficients.

Is A055606 a shifted version of this sequence?

Examples

			Row 5 of the triangle of q-binomial coefficients is [1, 1 + q + q^2 + q^3 + q^4, 1 + q + 2*q^2 + 2*q^3 + 2*q^4 + q^5 + q^6, 1 + q + 2*q^2 + 2*q^3 + 2*q^4 + q^5 + q^6, 1 + q + q^2 + q^3 + q^4, 1], so the max coefficient is 2. Hence a(5) = 2.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..200
• E. Friedman and M. Keith, Magic Carpets, J. Int Sequences, 3 (2000), #P.00.2.5.
• Eric W. Weisstein, q-Binomial Coefficient
• Wikipedia, q-binomial

Crossrefs

Cf. A002838, A022166, A029895, A055606, A076822.

Programs

Maple

f:= proc(n) local k, c, v, q;
  uses QDifferenceEquations;
  v:= 0:
  for k from 0 to n do
    c:= coeffs(expand(expand(QBinomial(n,k,q))),q);
    v:= max(v, max(c));
  od:
v
end proc:
map(f, [$0..50]); # Robert Israel, Oct 05 2016

Mathematica

Table[Coefficient[Expand[FunctionExpand[QBinomial[n, Floor[n/2], q]]], q, Floor[n^2/8]], {n, 0, 30}] (* Vladimir Reshetnikov, Sep 24 2021 *)

Formula

a(n) ~ sqrt(3) * 2^(n+2) / (Pi * n^2). - Vaclav Kotesovec, Oct 09 2016