Bouncing Factorial
Notation\(n\Lambda\)
TypeCombinatorial
Based onFactorial
Growth rate\(f_{2}(n)\)

The Bouncing Factorial is a type of factorial that multiplies together the integers from 1 to some number n, then back to 1, then back to (n-1), then to 1, then to (n-2), and so on. It is denoted \(n\Lambda\).

Example

For instance, the bouncing factorial of 9 is equal to 1×2×3×4×5×6×7×8×9×8×7×6×5×4×3×2×1×2×3×4×5×6×7×8×7×6×5×4×3×2×1×2×3×4×5×6×7×6×5×4×3×2×1×2×3×4×5×6×5×4×3×2×1×2×3×4×5×4×3×2×1×2×3×4×3×2×1×2×3×2×1×2×1, or 9,278,496,603,801,318,870,491,332,608,000,000,000.

Pictured above is a visualization of the bouncing factorial of 9, where every new multiplication peak has been colored for clarity. The numbers above the peaks refer to their height in units. The x-axis may be thought of as time, and the y-axis as quantity. The bouncing factorial of 9 is equal to the product of the quantities of all those little colored squares.

Values

The first few values of the function are:

• \(1Λ = 1\)
• \(2Λ = 2\) (1×2×1)
• \(3Λ = 24\) (1×2×3×2×1×2×1)
• \(4Λ = 3,456\) (1×2×3×4×3×2×1×2×3×2×1×2×1)
• \(5Λ = 9,953,280\) (1×2×3×4×5×4×3×2×1×2×3×4×3×2×1×2×3×2×1×2×1)
• \(6Λ = 859,963,392,000\) (1×2×3×4×5×6×5×4×3×2×1×...)
• \(7Λ = 3,120,635,156,889,600,000\) (1×2×3×4×5×6×7×6×5×4×3×2×1×...)
• \(8Λ = 634,153,008,009,974,906,880,000,000\) (1×2×3×4×5×6×7×8×7×6×5×4×3×2×1×...)
• \(9Λ = 9,278,496,603,801,318,870,491,332,608,000,000,000\) (1×2×3×4×5×6×7×8×9×8×7×6×5×4×3×2×1×...)
• \(10Λ = 12,218,100,099,725,239,100,847,669,366,019,325,952,000,000,000,000\) (1×2×3×4×5×6×7×8×9×10×9×8×7×6×5×4×3×2×1×...)

Formal description

The Bouncing Factorial of n can be formally defined as \(n(\prod_{i=1}^{n-1} i^{2n-2i+1})\). This formula holds true for all values of n greater than 1. When n equals one, the bouncing factorial is 1. It may also be recursively defined as \(Z_{n+1}={ {(n+1)!^2}/(n+1)}*Z_n\) where \(Z_1=1\).

It can also be calculated as n!×(n-1)!×(n-1)Λ for n > 1.

Primes

\(3\Lambda -1\) is prime. Because 3Λ is 24, and 24-1 is 23, and 23 is prime. As of the time this was written, no other primes have been found of the form \(n\Lambda -1\) for values of n less than or equal to 18.