Proof step e-34a8fe142a1eb34c

basis_map A013610 → concludes about basis_map_A013610 of Triangle T(n,k) = T(n-1,k) + 3*T(n-3,k-1) for k = 0..floor(n/3) 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": "A013610",
  "operation": "basis_map"
}

Premises

• Triangle T(n,k) = T(n-1,k) + 3*T(n-3,k-1) for k = 0..floor(n/3) 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 acbb74b7a183efe8027e87448bca23df9444d4fe0022adfbb46b7b94bc4cb819, 10,726 bytes (2,412 compressed); download JSON

Cited by

• n-1b544dd56a887d3b (real_rooted)
• n-1b544dd56a887d3b (one_signed_coefficients)
• n-1b544dd56a887d3b (hadamard_kernel)
• n-1b544dd56a887d3b (interlacing)