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 Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m).

Premises

Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m).
sequence: Every row P_n, n >= 1, 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-ddbb8c4f51a256ae · 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\":1,\"first_degree\":2,\"step\":1}}"
}

Claims

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

Certificate

preserver_image, SHA-256 73cd15a6a893806c91c4dbd38338d7d443143bc8e8925310dae188aa236b284e, 11,270 bytes (2,885 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-257453fbf28c3b7a (real_rooted)
n-257453fbf28c3b7a (one_signed_coefficients)
n-257453fbf28c3b7a (hadamard_kernel)