Proof step e-62a7be97cd0c2a0d

preserver_image affine, shift 1 → concludes about Triangle read by rows: A008277 * A007318. 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": 1,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"
}

Premises

• Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= 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": 0, "region": { "upper": { "closed": true, "value": "-1" } } }
• { "claim": "interlacing", "from_index": 1, "kind": "weak", "orientation": "unordered" }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: preserver_image, SHA-256 ba2acb5782a9e94ebe98c5c2b7e62caf41640a0e3a767f6fffd6b475d7a59032, 14,286 bytes (3,055 compressed); download JSON

Cited by

• n-bdae5aa3d1cac0ec (real_rooted)
• n-bdae5aa3d1cac0ec (one_signed_coefficients)
• n-bdae5aa3d1cac0ec (hadamard_kernel)
• n-bdae5aa3d1cac0ec (interlacing)