Proof step e-d6128f80e98dcc59

basis_map A007318 → concludes about basis_map_A007318 of Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (m*x + 2)*a(m+1) - a(… JSON

Theorem

Geometric basis map (elementary) polytool ?

Operation

Image under a basis map u^k -> P_k

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

Premises

• Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (m*x + 2)*a(m+1) - a(m… (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 addf5f3fc209d725a18c0460f2f7b3044593dcdc2074e66eeafb1ba247cadc18, 13,381 bytes (2,526 compressed); download JSON

Cited by

• n-4a57e8264fe984f4 (real_rooted)
• n-4a57e8264fe984f4 (one_signed_coefficients)
• n-4a57e8264fe984f4 (hadamard_kernel)
• n-4a57e8264fe984f4 (interlacing)