Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver reversal, shift 0

Concludes

reversal of shift_x_plus_one of Riordan array ((1+x)^3,x).

Premises

shift_x_plus_one of Riordan array ((1+x)^3,x).
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-26d21c6cbf3a9b4e · JSON

Operation

Image under a real-rootedness preserver

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

Claims

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

Certificate

preserver_image, SHA-256 1c9c30ff7e37a0d7204e34d57b499d5e360e1f085c2bb292b4a5492f1dbd3da6, 7,954 bytes (2,292 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-f5886681a2578968 (real_rooted)
n-f5886681a2578968 (one_signed_coefficients)
n-f5886681a2578968 (hadamard_kernel)