Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver reversal, shift 0

Concludes

reversal of theta_plus_one of a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.

Premises

derivative of basis-map coefficients of A scaled Hermite triangle.
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-f26705c57a06d249 · JSON

Operation

Image under a real-rootedness preserver

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

Claims

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

Certificate

preserver_image, SHA-256 57112ff0fa194ef06ef5d36f775baf6ce39d46ae94b9351200bed73fdafe0b45, 8,166 bytes (2,334 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-7adc12e1650df214 (one_signed_coefficients)
n-7adc12e1650df214 (hadamard_kernel)
n-7adc12e1650df214 (real_rooted)