Proof step e-ef5805120442c2b5

preserver_image affine, shift 0 → concludes about shift_x_plus_one of Euler's triangle: triangle of Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows. 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

• 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": 1, "region": { "upper": { "closed": true, "value": "-1" } } }
• { "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 9c2cdd46fc59d007a12c12345332962ca63da2daba1e0ee4fe1e51677d489165, 12,683 bytes (3,114 compressed); download JSON

Cited by

• n-d6d713aa669b74c1 (real_rooted)
• n-d6d713aa669b74c1 (one_signed_coefficients)
• n-d6d713aa669b74c1 (hadamard_kernel)
• n-d6d713aa669b74c1 (interlacing)