Proof step e-b7f464e4dcf953bf

preserver_image mobius, shift 0 → concludes about Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows (n >= 1, 1 <= 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

• eulerian (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 deef9b7db9b1b2e68023c33edc07f8403129cb891e7744464c6a6eaa4789dfac, 10,757 bytes (2,680 compressed); download JSON

Cited by

• n-d938c1937a93c4df (one_signed_coefficients)
• n-d938c1937a93c4df (hadamard_kernel)
• n-d938c1937a93c4df (real_rooted)