Proof step e-298e29fac11cdc8c

basis_map A007318 → concludes about basis_map_A007318 of Triangle read by rows: T(n,0) = 0 for n >= 0; T(n,2*k+1) = A152842(2*n,2*(n-k)) and T(n,2*k) = A152842(2*n,2*(n-k)+1) for n >= 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: T(n,0) = 0 for n >= 0; T(n,2*k+1) = A152842(2*n,2*(n-k)) and T(n,2*k) = A152842(2*n,2*(n-k)+1) for n >= 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" } } }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: basis_map, SHA-256 cb09d2f30065ed3a8cc9f32fd77d30f4b415d6c927cdbd72fe294047fd8add8d, 6,895 bytes (1,399 compressed); download JSON

Cited by

• n-1f565ad1ec7a715f (real_rooted)
• n-1f565ad1ec7a715f (one_signed_coefficients)
• n-1f565ad1ec7a715f (hadamard_kernel)