Proof step e-040c826eba4e7d74

preserver_image affine, shift 0 → concludes about Triangle read by rows: T(n, m) = S1(n, m)*10^(n-m), where S1 are the signed Stirling numbers of first kind A008275 (n >= 1, 1 <= m <= 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\":\"-1/5\",\"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": 0, "region": { "lower": { "closed": true, "value": "5" } } }
• { "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 8b05620c7a1e1566ec3f9f4f68cb703cfd7d714564f9ba122d8bde292ab33cfe, 10,415 bytes (2,924 compressed); download JSON

Cited by

• n-f45d29a8d57e6cc8 (real_rooted)
• n-f45d29a8d57e6cc8 (one_signed_coefficients)
• n-f45d29a8d57e6cc8 (interlacing)