Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver affine, shift 0

Concludes

basis_map_A007318 of Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).

Premises

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

Theorem

polytool ?

Lean

RealRooted Polynomial.Splits.comp_of_degree_le_one

Checked

polytool replayed · Lean not attempted

Technical details

Step id

e-c4464c8cda467541 · JSON

Operation

Image under a real-rootedness preserver

{
  "index_shift": 0,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"
}

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"
}

Certificate

preserver_image, SHA-256 4a6b3e865e24e3287409cc7829bc766608dae9ee73b65f09a323c6ab8cfeb2a6, 18,345 bytes (3,585 compressed); download

polytool replayed the certificate when the step was written.

Cited by

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