Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver reversal, shift 0

Concludes

shift_x_plus_one of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.

Premises

veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).
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-9385c38a718b9bf3 · JSON

Operation

Image under a real-rootedness preserver

{
  "index_shift": 0,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"reversal\",\"schedule\":{\"first_index\":1,\"first_degree\":1,\"step\":3}}"
}

Claims

{
  "claim": "real_rooted",
  "from_index": 1,
  "region": {
    "upper": {
      "closed": true,
      "value": "0"
    }
  }
}

Certificate

preserver_image, SHA-256 b8b8ee768280dd6bc9089b258d59282118a00246ca94695f696b53d28770fb74, 5,016 bytes (1,534 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-a6f555b0e6271a68 (one_signed_coefficients)
n-a6f555b0e6271a68 (hadamard_kernel)
n-a6f555b0e6271a68 (real_rooted)