Proof step e-97aa57c096c47b30

preserver_image affine, shift 0 → concludes about basis_map_A007318 of Triangle, read by rows, where T(n,k) = n!/(k!*(n-3*k)!*3^k) for n>=3*k>=0. JSON

Theorem

Affine, reversal and Möbius substitutions (elementary) polytool ?

Lean (RealRooted): Polynomial.Splits.comp_of_degree_le_one

Operation

Image under a real-rootedness preserver

{
  "index_shift": 0,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"
}

Premises

• Triangle, read by rows, where T(n,k) = n!/(k!*(n-3*k)!*3^k) for n>=3*k>=0. (sequence)
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]

Claims

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

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: preserver_image, SHA-256 0178e4e60a6e1cad62a84302fe19f95bb4d1c20305efea434f625c84d39f3961, 6,748 bytes (2,019 compressed); download JSON

Cited by

• n-b1d15686007b99d5 (one_signed_coefficients)
• n-b1d15686007b99d5 (hadamard_kernel)
• n-b1d15686007b99d5 (real_rooted)