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## On Claims Derived from Shifted Legendre Polynomial Coefficient Formulas

A few combinatorics arguments... perhaps you can think of some proofs? Hint: I did not approach this via induction proofs:

General Multiplicative Notions

Corollary

for $n \neq k$ and $n \neq k - 1$

Corollary

Corollary

Corollary: Binomial Multiplicative Formula

Corollary

Catalan Numbers

Corollary

Corollary

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