Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver reversal, shift 0

Concludes

shift_x_plus_one of Triangle related to the o.g.f.s. of the right-hand columns of A130534 (E(x,m=1,n)).

Premises

Triangle of Ward numbers T(n,k) read by rows.
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-a9ae2fd1b6b922e0 · 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 11d0e34a4d3250b7bf26a5edbb0da1b12cae411c417919e59d1c282d0bff378f, 15,851 bytes (3,083 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-c243a2756c4622a2 (real_rooted)
n-c243a2756c4622a2 (one_signed_coefficients)
n-c243a2756c4622a2 (hadamard_kernel)