binomial odd mode polynomial coefficients

is there a way to get these coefficients of the binomial polynomial, like shown in the photo bellow?

Thanks

[Walter Roberson]

That looks to be related to the Legendre polynomials, but not exactly the same; see https://en.wikipedia.org/wiki/Legendre_polynomials and notice P2(x) = 1/2 * (3*x^2 - 1) which would expand to (1.5*x^2 - 0.5) which looks too similar to (1.5x - 0.5x^3) for me to think there is no relationship.

The Legendre polynomials use that P notation.

The coefficients you show have no obvious relationship to Binomials or to bionomial coefficients

[John D'Errico]

The appearance of L on the left hand side is another subtle suggestion it MIGHT be Legendre related.

But really, the question to me is exactly what are you looking to do here? Do you have the polynomial, and wish to expand it? Or do you have that first relation, and wish to recognize it as the square of a polynomial of interest?

[Walter Roberson]

This appears to be continued at https://www.mathworks.com/matlabcentral/answers/342653-butterworth-maximal-flat-odd-polynomial