Proof step e-3c4e27dedb9c54a8

preserver_image affine, shift 0 → concludes about basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. 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

• Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. (sequence)
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4, 0]

Claims

• { "claim": "real_rooted", "from_index": 1, "region": { "lower": { "closed": false, "value": "-3" }, "upper": { "closed": true, "value": "1" } } }
• { "claim": "interlacing", "from_index": 2, "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 f3ecc14e84a19c598a911339469e2ba68b2103048575d8f0ed92c1d2f4509003, 19,477 bytes (3,797 compressed); download JSON

Cited by

• n-500943bf802dcc95 (real_rooted)
• n-500943bf802dcc95 (interlacing)