Proof step e-3c8d43f1dfa996d6

basis_map A007318 → concludes about basis_map_A007318 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": "A007318",
  "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 ce4378245615875252779721d09b46530c4129ef53310a527301da90cb43706a, 13,982 bytes (2,615 compressed); download JSON

Cited by

• n-83e32d0bc1c10d45 (one_signed_coefficients)
• n-83e32d0bc1c10d45 (hadamard_kernel)
• n-83e32d0bc1c10d45 (interlacing)
• n-83e32d0bc1c10d45 (real_rooted)