Proof step e-a938384cba6c4260

preserver_image affine, shift 0 → concludes about basis_map_A007318 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

Affine, reversal and Möbius substitutions (elementary) polytool ?

Lean (RealRooted): Polynomial.Splits.comp_of_degree_le_one

Operation

Image under a real-rootedness preserver

{
  "index_shift": 0,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"
}

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": "-1" } } }
• { "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: preserver_image, SHA-256 c26fed10557949bc309e527af2340dc8253f2714f31d2ba1eea8cbc68e6bc648, 21,301 bytes (5,735 compressed); download JSON

Cited by

• n-cca535f6115f5fc0 (one_signed_coefficients)
• n-cca535f6115f5fc0 (hadamard_kernel)
• n-cca535f6115f5fc0 (interlacing)
• n-cca535f6115f5fc0 (real_rooted)