Proof step e-b26ab3b12a5f2165

preserver_image affine, shift 0 → concludes about basis_map_A007318 of Triangle of coefficients in expansion of (1+3*x)^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 coefficients in expansion of (1+3*x)^n. (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 8fbe86d48bb7e20d11ddffaeb555ad705d666cd7e7e25a2b613a2f383c0bbc73, 9,155 bytes (2,633 compressed); download JSON

Cited by

• n-e8c6285b75cf03a2 (one_signed_coefficients)
• n-e8c6285b75cf03a2 (hadamard_kernel)
• n-e8c6285b75cf03a2 (real_rooted)
• n-e8c6285b75cf03a2 (interlacing)