Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver reversal, shift 0

Concludes

reversal of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).

Premises

Triangle read by rows: T(n, k) = binomial(n+k-1, 2*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-ce7da4444ad1ab27 · 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 a9e2d5c173db8afba716b4082c1eda12cb233a670216dad36c2c9c0ca003761b, 16,627 bytes (3,085 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-697c605ecd7e51b4 (real_rooted)
n-697c605ecd7e51b4 (one_signed_coefficients)
n-697c605ecd7e51b4 (hadamard_kernel)