Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver reversal, shift 0

Concludes

reversal of Triangle of scaled central factorial numbers, T(n,k) = A008958(n,n-k).

Premises

Triangle of scaled central factorial numbers, T(n,k) = A008958(n,n-k).
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-f98917cdc49d32c4 · 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 e96cbc5e5253aa2147ab829b3ee8f3f2e239557db0fb921c056d4df884c2cf8e, 7,377 bytes (1,990 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-d8236c7dd97f9760 (real_rooted)
n-d8236c7dd97f9760 (one_signed_coefficients)
n-d8236c7dd97f9760 (hadamard_kernel)