Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver mobius, shift 0

Concludes

basis_map_A013610 of Triangle of coefficients in expansion of (1+2*x)^n.

Premises

Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.
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-d821ce4bda1b2a22 · 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 5143d011577ef80e6290c9b3ec33dec149c2964e309873c2d02eb90fc106f1f9, 8,169 bytes (2,333 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-53ad300b8c449980 (one_signed_coefficients)
n-53ad300b8c449980 (hadamard_kernel)
n-53ad300b8c449980 (real_rooted)