A277218 - OEIS

%I #33 Sep 25 2021 04:16:11

%S 1,1,1,1,2,2,3,5,8,12,20,32,58,94,169,289,526,910,1667,2934,5448,9686,

%T 18084,32540,61108,110780,208960,381676,723354,1328980,2527074,

%U 4669367,8908546,16535154,31630390,58965214,113093022,211591218,406680465,763535450,1470597342,2769176514,5342750699,10089240974

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

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

%C Is A055606 a shifted version of this sequence?

%H Vaclav Kotesovec, <a href="/A277218/b277218.txt">Table of n, a(n) for n = 0..200</a>

%H E. Friedman and M. Keith, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/KEITH/carpet.html">Magic Carpets</a>, J. Int Sequences, 3 (2000), #P.00.2.5.

%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/q-BinomialCoefficient.html">q-Binomial Coefficient</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Q-binomial">q-binomial</a>

%F a(n) ~ sqrt(3) * 2^(n+2) / (Pi * n^2). - _Vaclav Kotesovec_, Oct 09 2016

%e 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.

%p f:= proc(n) local k, c, v, q;

%p   uses QDifferenceEquations;

%p   v:= 0:

%p   for k from 0 to n do

%p     c:= coeffs(expand(expand(QBinomial(n,k,q))),q);

%p     v:= max(v, max(c));

%p   od:

%p v

%p end proc:

%p map(f, [$0..50]); # _Robert Israel_, Oct 05 2016

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

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

%K nonn

%O 0,5

%A _Vladimir Reshetnikov_, Oct 05 2016