Proof step e-e8868415ed0fc739

basis_map A013611 → concludes about basis_map_A013611 of 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. JSON

Theorem

Geometric basis map (elementary) polytool ?

Operation

Image under a basis map u^k -> P_k

{
  "map": "A013611",
  "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 59ffb96cdd1098a42382b5ed3c0a04fb22cadc8f5e6bb71cc171a3ff820dc536, 13,988 bytes (2,619 compressed); download JSON

Cited by

• n-333a80cbe99b35f1 (real_rooted)
• n-333a80cbe99b35f1 (one_signed_coefficients)
• n-333a80cbe99b35f1 (hadamard_kernel)
• n-333a80cbe99b35f1 (interlacing)