On estimating the sum of coefficient moduli in Bernstein polynomials on a symmetric interval

For Bernstein polynomials on the symmetric interval $[-1,1]$, a growth rate problem for the sum of coefficient moduli is considered. Used representations of the polynomials is indicated. We give a possible way to solve the problem through special numerical objects (Pascal's trapezoids). These objects are related to various combinatorial identities. The obtained result improves Roulier's previous estimate, which applies to the sum of coefficient moduli as the index of the Bernstein polynomial increases.