Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver reversal, shift 0

Concludes

Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).

Premises

Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
sequence: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]

Theorem

polytool ?

Lean

RealRooted route reversed RealRooted.X_pow_mul_reverse_eq_zero_or_splits, RealRooted.reflect_eq_zero_or_splits

Checked

polytool replayed · Lean not attempted

Technical details

Step id

e-ebf2cced9e5f3cf5 · JSON

Operation

Image under a real-rootedness preserver

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

Claims

{
  "claim": "real_rooted",
  "from_index": 0,
  "region": {
    "upper": {
      "closed": true,
      "value": "0"
    }
  }
}

Certificate

preserver_image, SHA-256 f877231e39d9e86cc874438c9d54b3bdd219921e199cd4a2b52159a87332451c, 22,372 bytes (5,568 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-d6b7687ba0ff59dd (one_signed_coefficients)
n-d6b7687ba0ff59dd (hadamard_kernel)
n-d6b7687ba0ff59dd (real_rooted)