Proof step e-e3a638b293d2df1c

preserver_image affine, shift 0 → concludes about Triangle T(n,k) = 3*T(n-1,k) + T(n-4,k-1) for k = 0..floor(n/4), with T(0,0) = 1 and T(n,k) = 0 for n or k < 0, read by rows. 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\":\"1/243\",\"intercept\":\"0\"}"
}

Premises

• Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) + 3 * T(n-4,k-1) for k = 0..floor(n/4); T(n,k)=0 for n or k < 0. (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": "0" } } }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: preserver_image, SHA-256 c1dd4a40d0bfbf4a0691ad50d0ea8fce3dd426bf04ef3c1061ed0ab7eecb28a7, 7,432 bytes (2,247 compressed); download JSON

Cited by

• n-11a69976f40ceb7e (real_rooted)
• n-11a69976f40ceb7e (one_signed_coefficients)
• n-11a69976f40ceb7e (hadamard_kernel)