Proof step e-10a26558e3d19067

preserver_image affine, shift 0 → concludes about shift_x_plus_one of Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows, T(n, k) for 0 <= k <= n. 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": 0,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"
}

Premises

• Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows, T(n, k) for 0 <= k <= n. (sequence)
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-1, 0]

Claims

• { "claim": "real_rooted", "from_index": 1, "region": { "lower": { "closed": false, "value": "-2" }, "upper": { "closed": true, "value": "-1" } } }
• { "claim": "interlacing", "from_index": 2, "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 1a3039d1df25a25df76a3708b5378325b333602b568e8627d4c160474b9cb626, 17,142 bytes (3,372 compressed); download JSON

Cited by

• n-eec36ee0e876d0f2 (real_rooted)
• n-eec36ee0e876d0f2 (one_signed_coefficients)
• n-eec36ee0e876d0f2 (hadamard_kernel)
• n-eec36ee0e876d0f2 (interlacing)