Proof step e-fbd53d49ec356918

basis_map A007318 → concludes about shift_x_plus_one 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": "A007318",
  "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 6f5f2a56664e23996bfc2b198677988fc5b3ecff23ca2288497cb612f54c534f, 10,726 bytes (2,410 compressed); download JSON

Cited by

• n-b3808769b39a4e65 (one_signed_coefficients)
• n-b3808769b39a4e65 (hadamard_kernel)
• n-b3808769b39a4e65 (real_rooted)
• n-b3808769b39a4e65 (interlacing)