Proof step e-0c862e94735a3472

basis_map A007318 → concludes about basis_map_A007318 of Generalized Narayana numbers, T(n, k) = 3/(n + 1)*binomial(n + 1, k + 2)*binomial(n + 1, k - 1), triangular array read by rows. JSON

Theorem

Geometric basis map (elementary) polytool ?

Operation

Image under a basis map u^k -> P_k

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

Premises

• Generalized Narayana numbers, T(n, k) = 3/(n + 1)*binomial(n + 1, k + 2)*binomial(n + 1, k - 1), triangular array read by rows. (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 12cc83e917ef266d895b6bfcb61f749b2a8c225804d9215b9059ca0a20ccfd45, 14,251 bytes (2,667 compressed); download JSON

Cited by

• n-4355c823139108ab (real_rooted)
• n-4355c823139108ab (one_signed_coefficients)
• n-4355c823139108ab (hadamard_kernel)
• n-4355c823139108ab (interlacing)