Proof step e-41f01d4be10aeae3

preserver_image affine, shift 2 → concludes about Triangle read by rows: T(n, m) = S1(n, m)*4^(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": 2,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"affine\",\"slope\":\"-1/2\",\"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 >= 2, is zero or real-rooted with all roots in (-inf, 0]

Claims

• { "claim": "real_rooted", "from_index": 0, "region": { "lower": { "closed": true, "value": "2" } } }
• { "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 8b0947d552d466d5d99a2dbab78fd3358ba9050238c6d7e95ed255f2b99ce703, 10,426 bytes (2,942 compressed); download JSON

Cited by

• n-552ecb7379159c73 (real_rooted)
• n-552ecb7379159c73 (one_signed_coefficients)
• n-552ecb7379159c73 (interlacing)