Stirling Numbers of the First Kind Table (Unsigned)
Enter a maximum order N to generate the full triangle of unsigned Stirling numbers of the first kind c(n, k) from n=0 to N. Each row sums to n factorial.
Input
Enter a maximum order N to generate the triangle of unsigned Stirling numbers of the first kind c(n, k) from n=0 to N.
Integer from 0 to 12
Result
Sum of the n = 6 row
720
This equals 6 factorial (6!).
Triangle of unsigned Stirling numbers of the first kind
| row n / col k | k=0 | k=1 | k=2 | k=3 | k=4 | k=5 | k=6 |
|---|---|---|---|---|---|---|---|
| n=0 | 1 | — | — | — | — | — | — |
| n=1 | 0 | 1 | — | — | — | — | — |
| n=2 | 0 | 1 | 1 | — | — | — | — |
| n=3 | 0 | 2 | 3 | 1 | — | — | — |
| n=4 | 0 | 6 | 11 | 6 | 1 | — | — |
| n=5 | 0 | 24 | 50 | 35 | 10 | 1 | — |
| n=6 | 0 | 120 | 274 | 225 | 85 | 15 | 1 |
Each cell is c(n, k). Entries where k exceeds n are left blank since they are undefined.
Maximum order N
6
Sum of the n = 6 row
720
How it works
- The unsigned Stirling number of the first kind c(n, k) counts the ways to arrange n elements into k disjoint cycles.
- It satisfies the recurrence c(n, k) = c(n-1, k-1) + (n-1) * c(n-1, k), with c(0, 0) = 1 and zero outside the valid range.
- Each row sum Sum over k of c(n, k) equals n factorial, which serves as a useful check.
- The signed Stirling number of the first kind is s(n, k) = (-1)^(n-k) * c(n, k); this tool shows the unsigned values.
- All values are computed and displayed with arbitrary precision integers (BigInt) so large entries appear without rounding error.
- The lower triangular table leaves entries with k greater than n blank, since they are not defined.
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Stirling Numbers of the First Kind Table (Unsigned)