curl --request POST \
  --url https://api.iotools.cloud/v1/tool/stirling-numbers-calculator \
  --header 'Authorization: Bearer <token>' \
  --header 'Content-Type: application/json' \
  --data '
{
  "n": "6",
  "k": "3"
}
'

{
  "tool": "stirling-numbers-calculator",
  "tool_version": "1.0.2",
  "outputs": {
    "firstKind": "225",
    "secondKind": "90",
    "triangleRows": [
      {
        "n": "0",
        "c-n-0-n-first-kind": "1",
        "s-n-0-n-second-kind": "1"
      },
      {
        "n": "1",
        "c-n-0-n-first-kind": "0, 1",
        "s-n-0-n-second-kind": "0, 1"
      },
      {
        "n": "2",
        "c-n-0-n-first-kind": "0, 1, 1",
        "s-n-0-n-second-kind": "0, 1, 1"
      },
      {
        "n": "3",
        "c-n-0-n-first-kind": "0, 2, 3, 1",
        "s-n-0-n-second-kind": "0, 1, 3, 1"
      },
      {
        "n": "4",
        "c-n-0-n-first-kind": "0, 6, 11, 6, 1",
        "s-n-0-n-second-kind": "0, 1, 7, 6, 1"
      },
      {
        "n": "5",
        "c-n-0-n-first-kind": "0, 24, 50, 35, 10, 1",
        "s-n-0-n-second-kind": "0, 1, 15, 25, 10, 1"
      },
      {
        "n": "6",
        "c-n-0-n-first-kind": "0, 120, 274, 225, 85, 15, 1",
        "s-n-0-n-second-kind": "0, 1, 31, 90, 65, 15, 1"
      }
    ],
    "steps": "c(n, k) = c(n-1, k-1) + (n-1)·c(n-1, k)   [unsigned Stirling number of the first kind]\nS(n, k) = S(n-1, k-1) +    k·S(n-1, k)    [Stirling number of the second kind]\nBase case: c(0, 0) = S(0, 0) = 1\n\nc(6, 3) = 225 — permutations of 6 elements with exactly 3 cycles\nS(6, 3) = 90 — ways to partition a 6-element set into 3 non-empty subsets\n\nTriangle rows 0..6 (row i lists c(i,0)..c(i,i) and S(i,0)..S(i,i)):\n  n=0:  c = 1  |  S = 1\n  n=1:  c = 0, 1  |  S = 0, 1\n  n=2:  c = 0, 1, 1  |  S = 0, 1, 1\n  n=3:  c = 0, 2, 3, 1  |  S = 0, 1, 3, 1\n  n=4:  c = 0, 6, 11, 6, 1  |  S = 0, 1, 7, 6, 1\n  n=5:  c = 0, 24, 50, 35, 10, 1  |  S = 0, 1, 15, 25, 10, 1\n  n=6:  c = 0, 120, 274, 225, 85, 15, 1  |  S = 0, 1, 31, 90, 65, 15, 1"
  },
  "credits_used": 5,
  "credits_remaining": null
}