Proof step e-46cbbc49f6ae8edc

preserver_image affine, shift 0 → concludes about T(n,k) = 2^k*binomial(n,2k+1), where 0 <= k <= floor((n-1)/2), n >= 1. 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\":\"1/2\",\"intercept\":\"-1/4\"}"
}

Premises

• Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one version of) Fibonacci po… (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": "1/2" } } }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: preserver_image, SHA-256 923014c635f5453158ad1f9c9ba4dd758946a4bb628641a2ace8675cf215ad13, 7,386 bytes (2,236 compressed); download JSON

Cited by

• n-286e213a8791bd45 (real_rooted)