Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver reversal, shift 1

Concludes

Triangle read by rows giving successive states of cellular automaton generated by "Rule 20" initiated with a single ON (black) cell.

Premises

An interpolation operator, companion to A133080.
sequence: Every row P_n, n >= 1, 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-e5594dfd8f34b1b6 · JSON

Operation

Image under a real-rootedness preserver

{
  "index_shift": 1,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":3}}"
}

Claims

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

Certificate

preserver_image, SHA-256 a131f5fd435ba24a31bf691fe6de6adbae32980fdecec2c7e6329e4c7eb83e34, 12,174 bytes (2,682 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-73781d44d680ea54 (real_rooted)
n-73781d44d680ea54 (one_signed_coefficients)
n-73781d44d680ea54 (hadamard_kernel)