Proof step e-7b35457a4f9e8cf8

preserver_image affine, shift 0 → concludes about shift_x_plus_one of Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C(n+1,2*k+1). 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 T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C(n+1,2*k+1). (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 f6b1772f197ff6cbdac204710944df0d946bd2cddf95db635618bfa8cc1599dd, 9,908 bytes (2,732 compressed); download JSON

Cited by

• n-e5bdf85b655e989a (real_rooted)
• n-e5bdf85b655e989a (one_signed_coefficients)
• n-e5bdf85b655e989a (hadamard_kernel)