Proof step e-a8f11950a8fb3d94

preserver_image affine, shift 0 → concludes about basis_map_A007318 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 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) = C(n+3,3) * C(n,k), 0 <= k <= n. (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" } } }
• { "claim": "interlacing", "from_index": 1, "kind": "weak", "orientation": "unordered" }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: preserver_image, SHA-256 a7851bb95f8a2e80c2f18e5a32e6d56f69bc44976ed3bd88c38779431a6904c3, 11,620 bytes (2,927 compressed); download JSON

Cited by

• n-5d91f73259c0b712 (one_signed_coefficients)
• n-5d91f73259c0b712 (hadamard_kernel)
• n-5d91f73259c0b712 (interlacing)
• n-5d91f73259c0b712 (real_rooted)