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) are the coefficients of Charlier polynomials: A046716 transposed, for 0 <= k <= n.

Premises

Coefficients of a special case of Poisson-Charlier polynomials.
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-88b4a0ac58b15c92 · 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 3b85b9cfa4974c8503f9118c1b155484a69f158f7aa70efe1e8d251ee4c9c3f5, 10,973 bytes (2,718 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-3272d5bc697b9cb4 (one_signed_coefficients)
n-3272d5bc697b9cb4 (hadamard_kernel)
n-3272d5bc697b9cb4 (real_rooted)