Proof step e-791d6802d1e7f8a6

preserver_image affine, shift 0 → concludes about Triangle read by rows: T(n,k) is the number of words over alphabet {0,1,2,3} having exactly k occurrences of the string 01, where n>=0 and k>=0. 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\":\"1/16\",\"intercept\":\"-1/16\"}"
}

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" } } }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: preserver_image, SHA-256 d7fc8d46c1ddc1ab210bb5e46002188c51decb4fe5cfdfb7e37c36d4057f3ebc, 7,398 bytes (2,231 compressed); download JSON

Cited by

• n-e3a9cbbad5cb7ab4 (real_rooted)