Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver reversal, shift 0

Concludes

Triangle read by rows: the reversed x = 1+q Narayana triangle at m=2.

Premises

Triangle T(n,k) read by rows: (1/n) * C(2n+k,k-1) * C(n,k); n, k >= 1.
sequence: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]

Theorem

polytool ?

Lean

RealRooted route reversed RealRooted.X_pow_mul_reverse_eq_zero_or_splits, RealRooted.reflect_eq_zero_or_splits

Checked

polytool replayed · Lean not attempted

Technical details

Step id

e-ed3c81e13a9adfa6 · JSON

Operation

Image under a real-rootedness preserver

{
  "index_shift": 0,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"reversal\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"
}

Claims

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

Certificate

preserver_image, SHA-256 98ea18e8c831b2e84ddb26066bba1004b411aef90e108a184d56c1af957cc573, 7,783 bytes (2,283 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-f00f6de2d4bbc6f5 (real_rooted)
n-f00f6de2d4bbc6f5 (one_signed_coefficients)
n-f00f6de2d4bbc6f5 (hadamard_kernel)