Proof step e-423483b3c7deec75

basis_map A013609 → concludes about basis_map_A013609 of Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2. JSON

Theorem

Geometric basis map (elementary) polytool ?

Operation

Image under a basis map u^k -> P_k

{
  "map": "A013609",
  "operation": "basis_map"
}

Premises

• Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2. (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 1275dda3a7de09d6db1f496a93437e35d2311c96470224dbb8884b3dccca5f34, 13,634 bytes (2,581 compressed); download JSON

Cited by

• n-b7873c90b2011b57 (one_signed_coefficients)
• n-b7873c90b2011b57 (hadamard_kernel)
• n-b7873c90b2011b57 (interlacing)
• n-b7873c90b2011b57 (real_rooted)