Proof step e-cee0a9f908e74de4

preserver_image affine, shift 0 → concludes about Triangle read by rows, T(n, k) = Sum_{i=0..n} L'(n, n-i) * binomial(i, k), for k = 0..n-1. 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

• Triangular array A066667 or A008297 unsigned and transposed. (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": { "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 681486ca75a76a2677ac68cdc670c156e530d69c017c859202308c81c3b5d732, 14,408 bytes (3,099 compressed); download JSON

Cited by

• n-eb51f16fedb3c293 (one_signed_coefficients)
• n-eb51f16fedb3c293 (hadamard_kernel)
• n-eb51f16fedb3c293 (interlacing)
• n-eb51f16fedb3c293 (real_rooted)