Proof step e-fb63394f9047a486

basis_map A007318 → concludes about basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1). shifted by x -> … JSON

Theorem

Geometric basis map (elementary) polytool ?

Operation

Image under a basis map u^k -> P_k

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

Premises

• basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + … (sequence)
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 1]

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 42e0016394c21881af94d210212862d15ce1af0a905dd65487a891383b9a265f, 14,643 bytes (2,741 compressed); download JSON

Cited by

• n-0e6d847331141ca6 (one_signed_coefficients)
• n-0e6d847331141ca6 (hadamard_kernel)
• n-0e6d847331141ca6 (real_rooted)
• n-0e6d847331141ca6 (interlacing)