Proof step e-762ccb2ce32a5929

preserver_image affine, shift 0 → concludes about basis_map_A007318 of Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0<=k<=n). 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: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0<=k<=n). (sequence)
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0)

Claims

• { "claim": "real_rooted", "from_index": 0, "region": { "lower": { "closed": false, "value": "-3" }, "upper": { "closed": false, "value": "-1" } } }
• { "claim": "interlacing", "from_index": 1, "kind": "weak", "orientation": "unordered" }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: preserver_image, SHA-256 885404e9ae3361b75fa82018b65becd4a26822c2269616655f0793f0800a9ec7, 18,244 bytes (3,611 compressed); download JSON

Cited by

• n-47b361d1e904fbeb (one_signed_coefficients)
• n-47b361d1e904fbeb (hadamard_kernel)
• n-47b361d1e904fbeb (interlacing)
• n-47b361d1e904fbeb (real_rooted)