Proof step e-f1629c32ad3d0109

preserver_image affine, shift 0 → concludes about basis_map_A007318 of Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))). 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

• Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))). (sequence)
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0)

Claims

• { "claim": "real_rooted", "from_index": 0, "region": { "lower": { "closed": false, "value": "-9" }, "upper": { "closed": false, "value": "-1" } } }
• { "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 450f8c9a8d29ae3f229f0cde5ceab149ff6c0c0c67b8e9099ad3c88eac7604b8, 18,210 bytes (3,549 compressed); download JSON

Cited by

• n-317113238d23620b (one_signed_coefficients)
• n-317113238d23620b (hadamard_kernel)
• n-317113238d23620b (interlacing)
• n-317113238d23620b (real_rooted)