# Factorial Sum
Factorial Sum is a Python library for efficient and exact computation of factorial sums
S(š¯‘›) = 1! + 2! + 3! + ... n!
using a novel recursive identity introduced by Abdelkrim Meziani (2025).
This approach improves numerical stability, reduces recursion depth, and provides better performance compared to naive methods.
## Installation
```bash
pip install factorial-sum
```
## Usage
```python
from factorial_sum import factorial_sum
# Compute the sum of the first 5 factorials
print(factorial_sum(5)) # Output: 153
```
## Features
- Efficient algorithm based on a new recursive identity:
S(n) = nĀ² * (n-2)! + S(n-3)
- Faster than naive summation for large n
- Safe for large n using iterative computation (no recursion errors).
- Mathematically exact with Pythonā€™s arbitrary-precision integers.
- Lightweight & dependency-free (only uses Python standard library)
- Includes caching for performance
- Fully tested with pytest
## Applications
- Combinatorics and permutation analysis
- Graph enumeration problems
- Factorial number systems
- Teaching recursion and algorithm optimization
- Benchmarking and computational mathematics research
## Citation
If you use this package in academic work, please cite:
Abdelkrim Meziani (2025).
A Novel Recursive Identity for the Sum of Factorials with Computational and Combinatorial Applications.
[https://zenodo.org/records/16994879 / https://orcid.org/0009-0003-1849-4985]
## Author
**Abdelkrim Meziani**
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