Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver reversal, shift 1

Concludes

Row reversed version of triangle A201637 (second-order Eulerian triangle).

Premises

Triangle of second-order Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.
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-5f52c7cc32f7742d · JSON

Operation

Image under a real-rootedness preserver

{
  "index_shift": 1,
  "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 96ec117533de8dbdf469e25d3fa18c8a5e3be1dedb728970d4dd9892878af88f, 14,156 bytes (2,989 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-387e148a64b95986 (real_rooted)
n-387e148a64b95986 (one_signed_coefficients)
n-387e148a64b95986 (hadamard_kernel)