Proof step e-437a2dd956a9df17

preserver_image affine, shift 0 → concludes about basis_map_A007318 of T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle 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\",\"intercept\":\"1\"}"
}

Premises

• T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows. (sequence)
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-1, 0)

Claims

• { "claim": "real_rooted", "from_index": 1, "region": { "lower": { "closed": false, "value": "-2" }, "upper": { "closed": false, "value": "-1" } } }
• { "claim": "interlacing", "from_index": 2, "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 8ecba8238944457a93c4ec7e3d453705ab79780886e0d7416f9cf4d61a7fe4f5, 20,182 bytes (3,579 compressed); download JSON

Cited by

• n-0444a4f352052985 (hadamard_kernel)
• n-0444a4f352052985 (interlacing)
• n-0444a4f352052985 (real_rooted)
• n-0444a4f352052985 (one_signed_coefficients)