Proof step e-12ad66072da87843

preserver_image affine, shift 0 → concludes about basis-map coefficients of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1. 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\":\"-7\"}"
}

Premises

• Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1. (sequence)
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-5, -1]

Claims

• { "claim": "real_rooted", "from_index": 1, "region": { "lower": { "closed": false, "value": "2" }, "upper": { "closed": true, "value": "6" } } }
• { "claim": "interlacing", "from_index": 2, "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 1cf9ef616216ba83c98cc31c83874169bf8e7d1f9bce40be02745c1686165343, 19,637 bytes (3,905 compressed); download JSON

Cited by

• n-50a9a09f321fcea8 (real_rooted)
• n-50a9a09f321fcea8 (one_signed_coefficients)
• n-50a9a09f321fcea8 (interlacing)