/**
* Axiom: x₍n₎ = ∑ (-1)ⁿ⁻ᵏ L(n,k) x⁽ᵏ⁾
* Falling factorial can be represented as the summation of Lah numbers and rising factorials
*
* @dataProvider dataProivderForLahNumbers
*/
public function testFallingFactorialAsLahNumberAndRisingFactorial(int $x, $n)
{
$x₍n₎ = Combinatorics::fallingFactorial($x, $n);
$∑⟮−1⟯ⁿ⁻ᵏL⟮n、k⟯x₍k₎ = 0;
for ($k = 1; $k <= $n; $k++) {
$⟮−1⟯ⁿ⁻ᵏ = (-1) ** ($n - $k);
$L⟮n、k⟯ = Combinatorics::lahNumber($n, $k);
$x⁽ᵏ⁾ = Combinatorics::risingFactorial($x, $k);
$∑⟮−1⟯ⁿ⁻ᵏL⟮n、k⟯x₍k₎ += $⟮−1⟯ⁿ⁻ᵏ * $L⟮n、k⟯ * $x⁽ᵏ⁾;
}
$this->assertEquals($x₍n₎, $∑⟮−1⟯ⁿ⁻ᵏL⟮n、k⟯x₍k₎);
}
public function testFallingFactorialExceptionNLessThanZero()
{
$this->setExpectedException('MathPHP\\Exception\\OutOfBoundsException');
Combinatorics::fallingFactorial(5, -1);
}
