Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver reversal, shift 0

Concludes

Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix with 2's on the diagonal and 1's elsewhere.

Premises

Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.
sequence: Every row P_n, n >= 0, 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-011ba5f60d1d8c5c · JSON

Operation

Image under a real-rootedness preserver

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

Claims

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

Certificate

preserver_image, SHA-256 a8efd26cdacd518d2be4a4bda16b25a6a8b0f6a9a73d6133745ab060f4b45420, 11,048 bytes (3,042 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-801895951aa20f29 (real_rooted)
n-801895951aa20f29 (one_signed_coefficients)
n-801895951aa20f29 (hadamard_kernel)