Proof step e-fc673ff7432a109d

basis_map A007318 → concludes about basis_map_A007318 of Triangle T(n,k) = 3*T(n-1,k) + T(n-4,k-1) for k = 0..floor(n/4), with T(0,0) = 1 and 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) = 3*T(n-1,k) + T(n-4,k-1) for k = 0..floor(n/4), with T(0,0) = 1 and 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" } } }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: basis_map, SHA-256 b192d1c573fb4f5143a8e12ff4a5aa2c34d567d0be72eeac035f875a7f978684, 11,385 bytes (2,077 compressed); download JSON

Cited by

• n-905d1fc5d3865cdd (real_rooted)
• n-905d1fc5d3865cdd (one_signed_coefficients)
• n-905d1fc5d3865cdd (hadamard_kernel)