Proof step e-b3424874909a70f6

preserver_image affine, shift 0 → concludes about shift_x_plus_one of Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. 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), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. (sequence)
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0)

Claims

• { "claim": "real_rooted", "from_index": 0, "region": { "lower": { "closed": false, "value": "-5" }, "upper": { "closed": false, "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 834a21ff39df29db25c3231c027a9923b0251bac5261ec86bc18091b4ffed66b, 19,432 bytes (3,631 compressed); download JSON

Cited by

• n-d0986a7acf60271a (real_rooted)
• n-d0986a7acf60271a (one_signed_coefficients)
• n-d0986a7acf60271a (hadamard_kernel)
• n-d0986a7acf60271a (interlacing)