Multifactorial Calculator
Result:
Calculate multifactorials including double factorials (n!!), triple factorials, and more. These advanced factorial variations appear in combinatorics, analysis, and special mathematical sequences.
What are Multifactorials?
Multifactorials are generalizations of factorials where you multiply by every k-th number. Double factorial (n!!) multiplies every 2nd number, triple factorial every 3rd, and so on.
Multifactorial Formula
n!^(k) = n × (n-k) × (n-2k) × ... down to 1 or kDouble factorial: n!! = n × (n-2) × (n-4) × ...Triple factorial: n!!! = n × (n-3) × (n-6) × ...
Common Multifactorial Types
Double Factorial (!!)
- Even: n!! = n × (n-2) × (n-4) × ... × 2
- Odd: n!! = n × (n-2) × (n-4) × ... × 1
- Examples: 5!! = 15, 6!! = 48
- Formula: n!! = n!/(n-1)!! for n > 0
- Base cases: 0!! = 1, (-1)!! = 1
Triple Factorial (!!!)
- Pattern: n!!! = n × (n-3) × (n-6) × ...
- Examples: 7!!! = 7 × 4 × 1 = 28
- Stop at: 1, 2, or 3 (whichever reached first)
- Less common: Appears in specialized contexts
- Base cases: 1!!! = 1, 2!!! = 2, 3!!! = 3
Higher Order
- Quadruple: n!!!! (every 4th number)
- General: n!^(k) for any k
- Rare usage: Mostly theoretical
- Research: Combinatorial applications
- Notation: Various symbols used
Step-by-Step Examples
Even Numbers:
- 4!! = 4 × 2 = 8
- 6!! = 6 × 4 × 2 = 48
- 8!! = 8 × 6 × 4 × 2 = 384
- 10!! = 10 × 8 × 6 × 4 × 2 = 3840
Odd Numbers:
- 3!! = 3 × 1 = 3
- 5!! = 5 × 3 × 1 = 15
- 7!! = 7 × 5 × 3 × 1 = 105
- 9!! = 9 × 7 × 5 × 3 × 1 = 945
Pattern: Even double factorials grow much faster than odd ones.
Triple factorial calculations:
- 6!!! = 6 × 3 = 18
- 7!!! = 7 × 4 × 1 = 28
- 8!!! = 8 × 5 × 2 = 80
- 9!!! = 9 × 6 × 3 = 162
- 10!!! = 10 × 7 × 4 × 1 = 280
Note: The sequence depends on n mod 3 (remainder when divided by 3).
Double factorials have an interesting relationship to regular factorials:
For even n: n!! = 2^(n/2) × (n/2)!
For odd n: n!! = n! / (n-1)!!
Examples:
- 6!! = 48 and 2³ × 3! = 8 × 6 = 48 ✓
- 5!! = 15 and 5! / 4!! = 120 / 8 = 15 ✓
Common Values Table
| n | n! | n!! | n!!! | n!!!! |
|---|---|---|---|---|
| 0 | 1 | 1 | - | - |
| 1 | 1 | 1 | 1 | 1 |
| 2 | 2 | 2 | 2 | 2 |
| 3 | 6 | 3 | 3 | 3 |
| 4 | 24 | 8 | 4 | 4 |
| 5 | 120 | 15 | 10 | 5 |
| 6 | 720 | 48 | 18 | 12 |
| 7 | 5040 | 105 | 28 | 21 |
| 8 | 40320 | 384 | 80 | 32 |
Applications of Multifactorials
Mathematical Analysis
- Gamma function generalizations
- Special function definitions
- Series expansions
- Asymptotic analysis
- Hypergeometric functions
Combinatorics
- Specialized counting problems
- Restricted permutations
- Catalan number formulas
- Lattice path counting
- Partition functions
Physics & Engineering
- Quantum field theory calculations
- Statistical mechanics
- Signal processing
- Wave function normalizations
- Feynman diagram computations
Special Properties
📊 Mathematical Properties
Double Factorial Properties:
- Recurrence: n!! = n × (n-2)!!
- Even formula: (2n)!! = 2ⁿ × n!
- Odd formula: (2n-1)!! = (2n)! / (2ⁿ × n!)
- Product identity: n! = n!! × (n-1)!!
- Stirling approx: n!! ∼ √(πn/2) × (n/e)^(n/2)
General Properties:
- Growth rate: Much slower than regular factorial
- Parity patterns: Depends on n mod k
- Gamma function: Related to Γ(n/k + 1)
- Generating functions: Special forms exist
- Asymptotic behavior: Well-studied for large n
Frequently Asked Questions (FAQ)
These are completely different operations:
- n!! (double factorial): Multiply every 2nd number: 5!! = 5×3×1 = 15
- (n!)! (factorial of factorial): First calculate n!, then factorial of that: (5!)! = 120! (huge number)
- Notation importance: Position of parentheses matters critically
- Growth comparison: (n!)! grows astronomically faster than n!!
- Common confusion: Always check the notation carefully
Multifactorials appear in several specialized contexts:
- Combinatorics: Counting arrangements with specific restrictions
- Probability: Certain discrete probability distributions
- Mathematical analysis: Series expansions and special functions
- Physics: Quantum mechanics and statistical mechanics calculations
- Number theory: Studying factorial-like sequences
- Research mathematics: Various advanced applications
For large values, use these strategies:
- Logarithmic calculation: Work with log(n!^(k)) to avoid overflow
- Asymptotic approximations: Use Stirling-type formulas
- Modular arithmetic: If you only need the result modulo some number
- Specialized software: Mathematical programs with arbitrary precision
- Recurrence relations: Build up using smaller values
- Gamma function: Use Γ function relationships for non-integers
Base cases depend on the step size k:
- Double factorial: 0!! = 1, (-1)!! = 1 by convention
- For any k: Numbers 1 through k evaluate to themselves
- Examples: 1!!! = 1, 2!!! = 2, 3!!! = 3
- Zero case: 0!^(k) = 1 for all k (by convention)
- Negative cases: Usually undefined, except (-1)!! = 1
Yes, several useful relationships exist:
- Product identity: n! = n!! × (n-1)!! for all n
- Even double factorial: (2n)!! = 2ⁿ × n!
- Odd double factorial: (2n-1)!! = (2n)! / (2ⁿ × n!)
- General formula: n!^(k) relates to Γ((n+k)/k) in some cases
- Ratio formulas: Various relationships for specific values
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