Proof step e-2c04e0d67d228006

preserver_image affine, shift 0 → concludes about shift_x_plus_one of Triangle read by rows: T(n, k) = (n-k+1)*(k+1)*binomial(n, k). 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) = (n-k+1)*(k+1)*binomial(n, k). (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 f8f9346939451d76641202dcde6f0b8a5a8d19a30ae7b12c08ac5a87feeb68f5, 34,419 bytes (9,646 compressed); download JSON

Cited by

• n-aff30617856e629d (real_rooted)
• n-aff30617856e629d (one_signed_coefficients)
• n-aff30617856e629d (hadamard_kernel)