Proof step e-5e33c96e4e906d25

preserver_image affine, shift 0 → concludes about Triangle read by rows: T(n,k) = ((n+1)(n+2)/2) * binomial(n,k) (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": 0,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"
}

Premises

• Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2. (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": { "upper": { "closed": true, "value": "-1" } } }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: preserver_image, SHA-256 5813fa0ebedd178f0b1874da08244ef9ccd2d60273d7ae846a7ba3dae3c82e29, 7,240 bytes (1,981 compressed); download JSON

Cited by

• n-7f1b09730105c8f0 (one_signed_coefficients)
• n-7f1b09730105c8f0 (hadamard_kernel)
• n-7f1b09730105c8f0 (real_rooted)