Proof step e-3091186ded9433db

basis_map A007318 → concludes about basis_map_A007318 of Triangle T(n,k) = T(n-1,k) + 3*T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows. JSON

Theorem

Geometric basis map (elementary) polytool ?

Operation

Image under a basis map u^k -> P_k

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

Premises

• Triangle T(n,k) = T(n-1,k) + 3*T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows. (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 17b18bd98c2435c4144b62a9e2493f68db2ee68bff56d415afe40161aef77936, 10,696 bytes (2,405 compressed); download JSON

Cited by

• n-4d868f67dfcfb6b7 (real_rooted)
• n-4d868f67dfcfb6b7 (one_signed_coefficients)
• n-4d868f67dfcfb6b7 (hadamard_kernel)
• n-4d868f67dfcfb6b7 (interlacing)