Proof step e-da5b5b0d47f37943

basis_map A007318 → concludes about basis_map_A007318 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n. 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, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n. (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" } } }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: basis_map, SHA-256 62e0ee8326df5b6db95a2801eebb81948cc90b26c7b6fa46729bc30c3e1ffcd8, 6,875 bytes (1,388 compressed); download JSON

Cited by

• n-351cbe3ca6567d98 (real_rooted)
• n-351cbe3ca6567d98 (one_signed_coefficients)
• n-351cbe3ca6567d98 (hadamard_kernel)