PascalBinomial

The binomial coefficient, extended to a negative n so Pascal's identity still holds.

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PascalBinomial(n, m)n(n-1)…(n-m+1) / m!, for m >= 0 and any integer n

Typed signatures

Type	Package	Applies
(number, number) -> number	combinatorics	always

Domain: Collections

Details

• Agrees with Binomial where n >= m >= 0; for a negative n it is the standard generalized binomial coefficient, which still satisfies Pascal's recurrence P(n, m) = P(n-1, m-1) + P(n-1, m).
• A negative m is left unevaluated -- Wolfram's extension there is a documented gap, not yet cross-checked against a kernel.

Plot

Examples

1 more kept as data, checked against the oracles

Basic #

Agrees with Binomial(5, 2) in the standard range

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In$\mathrm{PascalBinomial}(5, 2)$Out$10$

Row $10$ of Pascal's triangle: $\binom{10}{3} = 120$

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In$\mathrm{PascalBinomial}(10, 3)$Out$120$

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In$\mathrm{PascalBinomial}(3, -1)$Out$0$

Scope #

C(-1, k) = (-1)^k, the standard identity for a negative upper index

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In$\mathrm{PascalBinomial}(-1, 3)$Out$-1$

The central entry of row $8$, $\binom84 = 70$

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In$\mathrm{PascalBinomial}(8, 4)$Out$70$

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In$N(\mathrm{PascalBinomial}(\frac{7}{3}, \frac{1}{5}), 20)$Out$1.333\,125\,424\,465\,028\,652\,2$

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In$\mathrm{MatrixFunction}(\operatorname{\_1}↦\mathrm{PascalBinomial}(\operatorname{\_1}, 2), \bigl\lbrack\bigl\lbrack\frac{5π}{6}, 0\bigr\rbrack, \bigl\lbrack\frac{3π}{2}, -\frac{π}{2}\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\frac{1}{72}×5π(5π-6), 0\bigr\rbrack, \bigl\lbrack\frac{1}{64}×5π(5π-6)-\frac{1}{64}×9π(2+π), \frac{π(2+π)}{8}\bigr\rbrack\bigr\rbrack$

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In$\mathrm{PascalBinomial}(\frac{9}{2}, \frac{7}{2})$Out$\frac{9}{2}$

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In$\mathrm{PascalBinomial}(1+i, 5)$Out$-\frac{1}{12}-\frac{1}{12}i$

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In$\mathrm{PascalBinomial}(n, 4)$Out$\frac{1}{24}n×(n-3)(n-2)(n-1)$

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In$\mathrm{PascalBinomial}(n, 0)$Out$1$

About

Examples	12
Checked against	Wolfram Language 12 agree
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