Proof step e-57079e0e5fed1718

preserver_image mobius, shift 0 → concludes about Triangle of coefficients in expansion of (2+3x)^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\":0,\"first_degree\":0,\"step\":1}}"
}

Premises

• Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j). (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": { "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 46d8a4867d9190fbefbd127a9b693da6e4eda34b89c010eab9378df4913fc063, 8,165 bytes (2,337 compressed); download JSON

Cited by

• n-5733d79f45a14084 (one_signed_coefficients)
• n-5733d79f45a14084 (hadamard_kernel)
• n-5733d79f45a14084 (real_rooted)