Proof step e-51c23bff4d2cb648

basis_map A013610 → concludes about basis_map_A013610 of 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 >= … JSON

Theorem

Geometric basis map (elementary) polytool ?

Operation

Image under a basis map u^k -> P_k

{
  "map": "A013610",
  "operation": "basis_map"
}

Premises

• 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… (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 5aa5345ba4ecd5720af5dd5af7f04ab767ddb29ded7ea28c99ef2a915277749c, 13,982 bytes (2,625 compressed); download JSON

Cited by

• n-cbba416b5dae7a51 (real_rooted)
• n-cbba416b5dae7a51 (one_signed_coefficients)
• n-cbba416b5dae7a51 (hadamard_kernel)
• n-cbba416b5dae7a51 (interlacing)