Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver mobius, shift 0

Concludes

theta_plus_one of a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.

Premises

theta_plus_one of a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
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-162b86bf556e8427 · 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 f46295c3f48d73b4566a4a7e160c14af3a0bf83d99cb761d38e88464e4843c56, 8,259 bytes (2,333 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-7b3743413970d91c (one_signed_coefficients)
n-7b3743413970d91c (hadamard_kernel)
n-7b3743413970d91c (real_rooted)