Proof step e-367cc697b56efa4f

basis_map A013610 → concludes about basis_map_A013610 of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!). 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) = RisingFactorial(n + 1, n) / (k! * (n - k)!). (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 8272f6bda3b4616b9fbe9bc16846f107d1155f2136a17aad921e9d6b67dcaa79, 13,217 bytes (2,516 compressed); download JSON

Cited by

• n-49350b617f7163d1 (real_rooted)
• n-49350b617f7163d1 (one_signed_coefficients)
• n-49350b617f7163d1 (hadamard_kernel)
• n-49350b617f7163d1 (interlacing)