Proof step e-b80b7af82ad2cf71

preserver_image affine, shift 0 → concludes about shift_x_plus_one of Triangle read by rows, 0 <= k <= n: T(n,k) = binomial(n-[(k+1)/2],[k/2])*(-1)^[(k+1)/2]. 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 read by rows, 0 <= k <= n: T(n,k) = binomial(n-[(k+1)/2],[k/2])*(-1)^[(k+1)/2]. (sequence)
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)

Claims

• { "claim": "real_rooted", "from_index": 1, "region": {} }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: preserver_image, SHA-256 4ff02ec9bc94052c0075832af5e603082518ec7f7666d4c98daed12dd8b2ebb1, 12,905 bytes (2,725 compressed); download JSON

Cited by

• n-e15113e58c389f73 (real_rooted)