Proof step e-afbdda6cf256c7ee

preserver_image affine, shift 0 → concludes about Triangle read by rows, 0 <= k <= n, T(n,k) = Sum_{j=0..n} A(n,j)*binomial(n-j,k) where A(n,j) are the Eulerian numbers A173018. 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 of Eulerian numbers T(n,k), 0 <= k <= n, read by rows. (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 f9c628f5fd8e75db3287bcb638745d32055d19dc9a1872b617ff521084434273, 7,285 bytes (2,192 compressed); download JSON

Cited by

• n-45ec3349543b0d37 (one_signed_coefficients)
• n-45ec3349543b0d37 (hadamard_kernel)
• n-45ec3349543b0d37 (real_rooted)