Proof step e-7fc1b55fb0aa373a

preserver_image affine, shift 1 → concludes about Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments. 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": 1,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"2\"}"
}

Premises

• Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one version of) Fibonacci po… (sequence)
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]

Claims

• { "claim": "real_rooted", "from_index": 0, "region": { "upper": { "closed": true, "value": "-2" } } }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: preserver_image, SHA-256 6878ecc528529237fe7a5c408fe99e97535f7cea37bbdb1560f9f1314f0e0a62, 6,706 bytes (2,049 compressed); download JSON

Cited by

• n-ac04e52510a48bb5 (real_rooted)
• n-ac04e52510a48bb5 (one_signed_coefficients)
• n-ac04e52510a48bb5 (hadamard_kernel)