Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver affine, shift 0

Concludes

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

Premises

basis_map_A007318 of 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 (-9, -1)

Theorem

polytool ?

Lean

RealRooted Polynomial.Splits.comp_of_degree_le_one

Checked

polytool replayed · Lean not attempted

Technical details

Step id

e-bb5180bc3cd5aafc · 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": "-10"
    },
    "upper": {
      "closed": false,
      "value": "-2"
    }
  }
}

{
  "claim": "interlacing",
  "from_index": 1,
  "kind": "weak",
  "orientation": "unordered"
}

Certificate

preserver_image, SHA-256 052e3789bbd8b60e0027fbd6c095706e5abcf8e9eec8a3826b02bfafcb84e7fc, 18,473 bytes (3,671 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-c77c01e985dfc77d (real_rooted)
n-c77c01e985dfc77d (one_signed_coefficients)
n-c77c01e985dfc77d (hadamard_kernel)
n-c77c01e985dfc77d (interlacing)