Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver affine, shift 0

Concludes

basis_map_A007318 of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).

Premises

Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).
sequence: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4, 0)

Theorem

polytool ?

Lean

RealRooted Polynomial.Splits.comp_of_degree_le_one

Checked

polytool replayed · Lean not attempted

Technical details

Step id

e-05347468080deadb · JSON

Operation

Image under a real-rootedness preserver

{
  "index_shift": 0,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"
}

Claims

{
  "claim": "real_rooted",
  "from_index": 1,
  "region": {
    "lower": {
      "closed": false,
      "value": "-5"
    },
    "upper": {
      "closed": false,
      "value": "-1"
    }
  }
}

{
  "claim": "interlacing",
  "from_index": 2,
  "kind": "weak",
  "orientation": "unordered"
}

Certificate

preserver_image, SHA-256 9e7683e9d08514531efc4f3127edc25a041f67903c6e889c1bfafb854b3d38bd, 18,084 bytes (3,413 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-e7f9db5cff35db04 (one_signed_coefficients)
n-e7f9db5cff35db04 (hadamard_kernel)
n-e7f9db5cff35db04 (interlacing)
n-e7f9db5cff35db04 (real_rooted)