binomial odd mode polynomial coefficients

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is there a way to get these coefficients of the binomial polynomial, like shown in the photo bellow?
Thanks
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John D'Errico
John D'Errico 2017-5-29
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?

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