Proof step e-b8d37b5fe37403cf

preserver_image affine, shift 0 → concludes about Triangle T(n,k) = 3*T(n-1,k) + T(n-3,k-1) for k = 0..floor(n/3) 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\":\"1/27\",\"intercept\":\"0\"}"
}

Premises

• Triangle read by rows, formed from antidiagonals of the antidiagonals (A011973) of Pascal's triangle (A007318). (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 b629a2be12674c53ee6d3ee2dc3377c8e7fcbfa64e5bfc50be6aba22f4a5db33, 7,395 bytes (2,233 compressed); download JSON

Cited by

• n-a5b33928f20c14a2 (real_rooted)
• n-a5b33928f20c14a2 (one_signed_coefficients)
• n-a5b33928f20c14a2 (hadamard_kernel)