Proof step e-38a7985c4c1aa2e3

basis_map A007318 → concludes about basis_map_A007318 of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck … 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 double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck p… (sequence)
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]

Claims

• { "claim": "real_rooted", "from_index": 1, "region": { "upper": { "closed": true, "value": "0" } } }
• { "claim": "interlacing", "from_index": 2, "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 30e69d3c1d2a11957fc94ed2374bbfc3a847c8e9861841d75bc59aafd44504c1, 13,998 bytes (2,627 compressed); download JSON

Cited by

• n-6ad396ce174787e3 (real_rooted)
• n-6ad396ce174787e3 (one_signed_coefficients)
• n-6ad396ce174787e3 (hadamard_kernel)
• n-6ad396ce174787e3 (interlacing)