Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver mobius, shift 0

Concludes

mobius_x_over_x_plus_one of reversal of Triangle read by rows: A008277 * A007318.

Premises

reversal of Triangle read by rows: A008277 * A007318.
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-5b48309baef5916c · 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 aadbf62e907078ba8b79dc9f5fb9ac7bba1fb71f8e2eeb06cdb8d124a7ae39c5, 26,816 bytes (4,594 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-6c46082d3ce6c4b2 (real_rooted)
n-6c46082d3ce6c4b2 (one_signed_coefficients)
n-6c46082d3ce6c4b2 (hadamard_kernel)