Proof step e-7103e40cdd5e6d87

preserver_image mobius, shift 0 → concludes about mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) = binomial(n+1,k+1)*binomial(2*n-k+1,k)/(n+1), 0<=k<=n. JSON

Theorem

Affine, reversal and Möbius substitutions (elementary) polytool ?

Lean (RealRooted): route fTransform RealRooted.splits_fPolynomial

Operation

Image under a real-rootedness preserver

{
  "index_shift": 0,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"
}

Premises

• Triangle read by rows: T(n,k) = binomial(n+1,k+1)*binomial(2*n-k+1,k)/(n+1), 0<=k<=n. (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": 1, "region": { "lower": { "closed": true, "value": "-1" }, "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 6d53be7dfd9257b02ddcd476bce4bf70b307e18e5e99b46c95931ebb9d04ec79, 22,115 bytes (5,577 compressed); download JSON

Cited by

• n-48a9203d1b5d504a (real_rooted)
• n-48a9203d1b5d504a (one_signed_coefficients)
• n-48a9203d1b5d504a (hadamard_kernel)