Proof step e-9b9e29c7199cdb19

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

Claims

• { "claim": "real_rooted", "from_index": 1, "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 0aacd046fb497962650a53aae403fab0d2aa80e236eb4e397fbaf57b266da2e7, 20,926 bytes (5,283 compressed); download JSON

Cited by

• n-6ee126ce99a6028c (real_rooted)
• n-6ee126ce99a6028c (one_signed_coefficients)
• n-6ee126ce99a6028c (hadamard_kernel)