Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver reversal, shift 0

Concludes

Triangle read by rows: T(n, k) = binomial(2n, 2k).

Premises

Triangle read by rows: T(n, k) = binomial(2n, 2k).
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-f65b10cdc89a8c38 · JSON

Operation

Image under a real-rootedness preserver

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

Claims

{
  "claim": "real_rooted",
  "from_index": 1,
  "region": {
    "upper": {
      "closed": true,
      "value": "0"
    }
  }
}

Certificate

preserver_image, SHA-256 35fc224903a518c30ec2af312e593eb667dd48b06daae9b0ed37f5b8af6754eb, 13,591 bytes (3,132 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-f7935c36a3c0ec8c (one_signed_coefficients)
n-f7935c36a3c0ec8c (hadamard_kernel)
n-f7935c36a3c0ec8c (real_rooted)