Proof step e-6e459812af5dc78f

preserver_image affine, shift -1 → concludes about Triangle of Stirling numbers of first kind, s(n,k), n >= 0, 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": -1,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"affine\",\"slope\":\"-2\",\"intercept\":\"-1\"}"
}

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": "-1/2" } } }
• { "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 6481b35b66795aae2a298526943ecd4432c88b9759d7a483b0420d146f1149f8, 10,434 bytes (2,943 compressed); download JSON

Cited by

• n-a52b649d7348b7a5 (real_rooted)
• n-a52b649d7348b7a5 (interlacing)