Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver reversal, shift 0

Concludes

Triangle of odd-numbered terms in rows of Pascal's triangle.

Premises

Triangle read by rows: T(n,k) is the number of binary words of length n with k strictly increasing runs, for 0<=k<=n.
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-d44b19a7da2ce27a · 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 35754890e2d1b4f00637680e3bf2e44a92924e1459ae290c77d5ed436a346aff, 9,768 bytes (2,679 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-fd133d9d2402ed18 (one_signed_coefficients)
n-fd133d9d2402ed18 (hadamard_kernel)
n-fd133d9d2402ed18 (real_rooted)