Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver mobius, shift 0

Concludes

basis_map_A013611 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).

Premises

basis_map_A013609 of Twice Pascal's triangle A007318: T(n,k) = 2*C(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-590d07e01e21ee3b · 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 43ed9ea3170a9b1eeeaaa1306ca0da1666b65bc35e27f3b12925251d3bae07fc, 8,182 bytes (2,351 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-a67bbce5c2713a1e (one_signed_coefficients)
n-a67bbce5c2713a1e (hadamard_kernel)
n-a67bbce5c2713a1e (real_rooted)