Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver mobius, shift 0

Concludes

basis_map_A013609 of Triangle read by rows: row n consists of n zeros followed by n!.

Premises

Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).
sequence: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]

Theorem

polytool ?

Lean

RealRooted route fTransform RealRooted.splits_fPolynomial

Checked

polytool replayed · Lean not attempted

Technical details

Step id

e-e73829f53d419ea2 · JSON

Operation

Image under a real-rootedness preserver

{
  "index_shift": 0,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"
}

Claims

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

Certificate

preserver_image, SHA-256 d388a50e1dc96bfa7d32eb902c61c782e033242abfa25ec66e3bcce4e3faa702, 9,737 bytes (2,537 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-439ac61e3f209ac0 (one_signed_coefficients)
n-439ac61e3f209ac0 (hadamard_kernel)
n-439ac61e3f209ac0 (real_rooted)