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 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.

Premises

shift_x_plus_one of 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 (-2, -1]

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-203a3bddd606a435 · 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": {
    "lower": {
      "closed": true,
      "value": "-1"
    },
    "upper": {
      "closed": true,
      "value": "0"
    }
  }
}

Certificate

preserver_image, SHA-256 70abc8d5c300ab190b6ac5cc38c7237c7e5e124fcc6d4b5210e24748ee9fc367, 16,094 bytes (3,147 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-efc52f864f8006b6 (real_rooted)
n-efc52f864f8006b6 (one_signed_coefficients)
n-efc52f864f8006b6 (hadamard_kernel)