Proof step e-fb809a8e7c094d2b

preserver_image reversal, shift 1 → concludes about Triangle read by rows: T(n,k) = binomial(k,n-k). JSON

Theorem

Affine, reversal and Möbius substitutions (elementary) polytool ?

Lean (RealRooted): route reversed RealRooted.X_pow_mul_reverse_eq_zero_or_splits, RealRooted.reflect_eq_zero_or_splits

Operation

Image under a real-rootedness preserver

{
  "index_shift": 1,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"
}

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 >= 1, is zero or real-rooted with all roots in (-inf, 0]

Claims

• { "claim": "real_rooted", "from_index": 0, "region": { "upper": { "closed": true, "value": "0" } } }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: preserver_image, SHA-256 52efcb9cc125f7f1ed27984803142e867244ec75f09f208c2c667c1c8d0d7dd7, 6,896 bytes (2,080 compressed); download JSON

Cited by

• n-21854f4257effb01 (real_rooted)
• n-21854f4257effb01 (one_signed_coefficients)
• n-21854f4257effb01 (hadamard_kernel)