Proof step e-d7c9e146299b18aa

preserver_image affine, shift -1 → concludes about Triangle of coefficients in expansion of (x-1)*(x-3)*(x-5)*...*(x-(2*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": -1,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"affine\",\"slope\":\"-1\",\"intercept\":\"0\"}"
}

Premises

• Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x. (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": 1, "region": { "lower": { "closed": true, "value": "0" } } }
• { "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 82f136ed346c58ac93963f115640d8ec421eff5485d64e975f921b247ff42ad1, 10,398 bytes (2,920 compressed); download JSON

Cited by

• n-17f1d12052bd159d (real_rooted)
• n-17f1d12052bd159d (one_signed_coefficients)
• n-17f1d12052bd159d (interlacing)