Real-rooted atlas

Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver mobius, shift 0

Concludes
mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) = (4n-4k+1) * T(n-1,k-1) + (4k-3) * T(n-1,k).
Premises
Theorem
polytool ?
Lean
RealRooted route fTransform RealRooted.splits_fPolynomial
Checked
polytool replayed · Lean not attempted
Technical details
Step id
e-da76e5cf7ae9b6e5 · JSON
Operation
Image under a real-rootedness preserver
{
  "index_shift": 0,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"
}
Claims
{
  "claim": "real_rooted",
  "from_index": 1,
  "region": {
    "lower": {
      "closed": true,
      "value": "-1"
    },
    "upper": {
      "closed": true,
      "value": "0"
    }
  }
}
Certificate
preserver_image, SHA-256 1e12e8c6aa3e90ae43e11ec39bb3a405914a416655ba8a1c188e13015eb9eefe, 10,973 bytes (2,762 compressed); download
polytool replayed the certificate when the step was written.
Cited by