The binomial coefficient of two natural numbers with zero is the value at that pair of the unique map satisfying Pascal's recursion.
In the setting of The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion, let , and let be the map of Binomial Coefficients: Existence and Uniqueness by Pascal's Recursion, Vanishing above the Diagonal, Diagonal and First Values, the Factorial Formula and Symmetry §recursion.
The binomial coefficient is the element of .
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