Proof step e-5fe7e1b16eb84690

basis_map A007318 → concludes about Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks at level one (n >= 1; 0 <= k <= n-1). JSON

Theorem

Geometric basis map (elementary) polytool ?

Operation

Image under a basis map u^k -> P_k

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

Premises

• Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan triangle. (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 28d1b4ae5df1fe3d8facd7a0658af382cbd62f03cc99d8678d15cb7a206e419b, 13,984 bytes (2,608 compressed); download JSON

Cited by

• n-57b4b14604f4c0d7 (one_signed_coefficients)
• n-57b4b14604f4c0d7 (hadamard_kernel)
• n-57b4b14604f4c0d7 (real_rooted)
• n-57b4b14604f4c0d7 (interlacing)