Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver reversal, shift 0

Concludes

Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.

Premises

Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.
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-9228aa17d52d8242 · 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 8cbb1c2d689c8664ba8b8d0dd4e94485aa4a009a94756bf43dae07bb1cfd37f4, 9,514 bytes (2,503 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-75a6475b3e851dc5 (one_signed_coefficients)
n-75a6475b3e851dc5 (hadamard_kernel)
n-75a6475b3e851dc5 (real_rooted)