Proof step e-c201453b6387188f

basis_map A007318 → concludes about shift_x_plus_one of Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pieces of 1 X 1 tiles … JSON

Theorem

Geometric basis map (elementary) polytool ?

Operation

Image under a basis map u^k -> P_k

{
  "map": "A007318",
  "operation": "basis_map"
}

Premises

• Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pieces of 1 X 1 tiles … (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": "0" } } }
• { "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: basis_map, SHA-256 d8aef51fc448a63899748043fe125d12fd3d432a0bf8c8349f5c79d04aab7867, 10,696 bytes (2,405 compressed); download JSON

Cited by

• n-0d89b6177f6a1dbb (one_signed_coefficients)
• n-0d89b6177f6a1dbb (hadamard_kernel)
• n-0d89b6177f6a1dbb (real_rooted)
• n-0d89b6177f6a1dbb (interlacing)