Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver reversal, shift 0

Concludes

shift_x_plus_one of Euler's triangle: triangle of Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.

Premises

Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.
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-fed781e782380521 · 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 c4b460da64f030d37f7a1bcbe823b8fc6c72f6c21e694a1da61719a9bff15bc1, 15,848 bytes (3,103 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-d6d713aa669b74c1 (one_signed_coefficients)
n-d6d713aa669b74c1 (hadamard_kernel)
n-d6d713aa669b74c1 (real_rooted)