Proof step e-d0c14addeb793323

basis_map A007318 → concludes about Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pieces of 1 X 2 tiles (0 <= k <= floor(n/2)… JSON

Theorem

Geometric basis map (elementary) polytool ?

Operation

Image under a basis map u^k -> P_k

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

Premises

• Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one version of) Fibonacci po… (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 26ac1ed4706fcacf6748d0e4d0fde57744d00e74cf4c621345276aba647daf06, 10,694 bytes (2,400 compressed); download JSON

Cited by

• n-7d332fcb7bf9a0f7 (one_signed_coefficients)
• n-7d332fcb7bf9a0f7 (hadamard_kernel)
• n-7d332fcb7bf9a0f7 (real_rooted)
• n-7d332fcb7bf9a0f7 (interlacing)