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) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0<=k<=n).

Premises

Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0<=k<=n).
sequence: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0)

Theorem

polytool ?

Lean

RealRooted Polynomial.Splits.comp_of_degree_le_one

Checked

polytool replayed · Lean not attempted

Technical details

Step id

e-57dbfc196a975a77 · 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": 0,
  "region": {
    "lower": {
      "closed": false,
      "value": "-3"
    },
    "upper": {
      "closed": false,
      "value": "-1"
    }
  }
}

{
  "claim": "interlacing",
  "from_index": 1,
  "kind": "weak",
  "orientation": "unordered"
}

Certificate

preserver_image, SHA-256 cc4d9e5f953111ffaf40d23402218f5dbcdfa3370acc61382b437bcbf8a0bf6a, 18,379 bytes (3,647 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-47b361d1e904fbeb (hadamard_kernel)
n-47b361d1e904fbeb (interlacing)
n-47b361d1e904fbeb (real_rooted)
n-47b361d1e904fbeb (one_signed_coefficients)