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

0,5

Comments

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

Is A055606 a shifted version of this sequence?

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

Formula

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

Example

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.

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 *)

Crossrefs

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

Sequence in context: A018136 A243853 A293419 * A293643 A022863 A236393

Adjacent sequences: A277215 A277216 A277217 * A277219 A277220 A277221

Keyword

nonn

Author

Vladimir Reshetnikov, Oct 05 2016

Status

approved