Proof step e-61bc78adfe1eb25a

preserver_image reversal, shift 1 → concludes about Triangle of Eulerian numbers T(n,k), 0 <= k <= n, read by rows. JSON

Theorem

Affine, reversal and Möbius substitutions (elementary) polytool ?

Lean (RealRooted): route reversed RealRooted.X_pow_mul_reverse_eq_zero_or_splits, RealRooted.reflect_eq_zero_or_splits

Operation

Image under a real-rootedness preserver

{
  "index_shift": 1,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"
}

Premises

• Euler's triangle: triangle of Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows. (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": "0" } } }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: preserver_image, SHA-256 f94e731a1dcc548a51f8e006b07d09073788d409c762f103b10a8372d7780374, 10,865 bytes (2,875 compressed); download JSON

Cited by

• n-e24b36ff0794dfbd (real_rooted)
• n-e24b36ff0794dfbd (one_signed_coefficients)
• n-e24b36ff0794dfbd (hadamard_kernel)