Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver reversal, shift 0

Concludes

reversal of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1 <= k <= m*(n + 1)} and m = 3.

Premises

Triangle read by rows, the Bell transform of the triple factorial numbers A007559(n+1) without column 0.
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-83811250d0324fbe · 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 cc8ebfc98b21a9dee0e89089e17b0865617e1ff25036344c3a13ac9f2e83e7ee, 10,676 bytes (2,729 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-b0dcdb0a8a0902b9 (real_rooted)
n-b0dcdb0a8a0902b9 (one_signed_coefficients)
n-b0dcdb0a8a0902b9 (hadamard_kernel)