Proof step e-ecb3a914f88de7f3

preserver_image affine, shift 0 → concludes about basis_map_A007318 of Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2). 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, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2). (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 7b9a88055748da8c7a56b1af290e54e0bb2a34b2c7e80c322c777fd1e2e6d234, 11,656 bytes (2,940 compressed); download JSON

Cited by

• n-e93f0e54dbc33ad5 (one_signed_coefficients)
• n-e93f0e54dbc33ad5 (hadamard_kernel)
• n-e93f0e54dbc33ad5 (interlacing)
• n-e93f0e54dbc33ad5 (real_rooted)