Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver reversal, shift 0

Concludes

Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

Premises

Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).
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-bd75a21c4b77773f · 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 5d8740dbd1b72b9605897feee6a55c42e6b8bc4b4584f53550e7b422e5b9a814, 22,398 bytes (5,507 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-e7331b0d50084c15 (one_signed_coefficients)
n-e7331b0d50084c15 (hadamard_kernel)
n-e7331b0d50084c15 (real_rooted)