Proof step e-480f02718c8257df

preserver_image affine, shift 0 → concludes about shift_x_plus_one of Triangle read by rows: T(n, k) = T(n-1, k-1) - T(n-2, k), with T(n, n) = 1, T(n, n-1) = -2. JSON

Theorem

Affine, reversal and Möbius substitutions (elementary) polytool ?

Lean (RealRooted): Polynomial.Splits.comp_of_degree_le_one

Operation

Image under a real-rootedness preserver

{
  "index_shift": 0,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"
}

Premises

• Triangle read by rows: T(n, k) = T(n-1, k-1) - T(n-2, k), with T(n, n) = 1, T(n, n-1) = -2. (sequence)
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 3)

Claims

• { "claim": "real_rooted", "from_index": 0, "region": { "lower": { "closed": false, "value": "-3" }, "upper": { "closed": false, "value": "2" } } }
• { "claim": "interlacing", "from_index": 1, "kind": "weak", "orientation": "unordered" }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: preserver_image, SHA-256 72ea3ab20ae1f1f76b3282a701bab463c2cfb900777726d269fca2f8df0d15ce, 25,925 bytes (6,330 compressed); download JSON

Cited by

• n-366c29e67ccc8d94 (real_rooted)
• n-366c29e67ccc8d94 (interlacing)