Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver reversal, shift 0

Concludes

veronese_section_2_0 of Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).

Premises

Triangle read by rows: T(n,k) is the number of flattened Catalan words of length n with exactly k runs of descents.
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-8f2094047437dc33 · 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 9eb9b2655041404d2b3994da393d58ae7296974a7c3b2c75f26a55eee927d4f1, 9,742 bytes (2,603 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-b6912a345c4fb865 (real_rooted)
n-b6912a345c4fb865 (one_signed_coefficients)
n-b6912a345c4fb865 (hadamard_kernel)