Proof step e-29d505b34d506bd7

basis_map A007318 → concludes about basis_map_A007318 of Triangle, read by rows, where T(n,k) = n!/(k!*(n-3*k)!*3^k) for n>=3*k>=0. 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, where T(n,k) = n!/(k!*(n-3*k)!*3^k) for n>=3*k>=0. (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 452f82d8cdd8cf977fe223e1097788445c360f6f14859f3f5a5089584a5d5953, 10,683 bytes (2,354 compressed); download JSON

Cited by

• n-b1d15686007b99d5 (real_rooted)
• n-b1d15686007b99d5 (one_signed_coefficients)
• n-b1d15686007b99d5 (hadamard_kernel)
• n-b1d15686007b99d5 (interlacing)