Proof step e-071c62da6fa41f89

basis_map A007318 → concludes about shift_x_plus_one of Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) + 3 * T(n-4,k-1) for k = 0..floor(n/4); T(n,k)=0 for n or k < 0. 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(0,0) = 1; T(n,k) = T(n-1,k) + 3 * T(n-4,k-1) for k = 0..floor(n/4); T(n,k)=0 for n or 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": "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 6defff6fbab115b0bb9d88e3e1f1898c30c4e761d77c50de502ea3fc47c4b7f4, 10,756 bytes (2,411 compressed); download JSON

Cited by

• n-1afd26bf28e20fd9 (one_signed_coefficients)
• n-1afd26bf28e20fd9 (hadamard_kernel)
• n-1afd26bf28e20fd9 (real_rooted)
• n-1afd26bf28e20fd9 (interlacing)