The "Hockey-stick identity" is a mathematical identity in combinatorics that describes a certain relationship involving binomial coefficients. It gets its name from the hockey stick shape that graphs of the identity can resemble.
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For nonnegative integers , the binomial coefficients satisfyThe identity follows by mathematical induction: adding the next term combines two adjacent binomial coefficients through Pascal's identity. An equivalent form is , a diagonal sum in Pascal's triangle. It provides closed forms for sums of binomial coefficients without expanding individual factorials.