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 theta_plus_one of Number triangle T(n,k)=C(n,k)C(2,n-k).

Premises

theta_plus_one of Number triangle T(n,k)=C(n,k)C(2,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-88cc6607d2d2464c · 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 ff47e4db82f0202791c954fce5d1fdad53644412eac8d641cc32506da6db29e9, 8,503 bytes (2,066 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-2ceed87b380f70ef (real_rooted)
n-2ceed87b380f70ef (one_signed_coefficients)
n-2ceed87b380f70ef (hadamard_kernel)