Proof step

Liu–Wang theorem for three-term recurrences

Theorem check of the sequence's own rule liu wang three term

Concludes: T(n, k) = binomial(n-k-1, k-1)*(n-k)!/k! for n >= 0 and 0 <= k <= floor(n/2). Irregular triangle read by rows.
Premises: none: a direct theorem check
Theorem: [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ?
Lean: RealRooted RealRooted.MaWang.strictInterl_of_interlaces_evalCoeff_nonpos
Checked: polytool replayed · Lean not attempted

Technical details

Step id

e-cb450d96e5764cbc · JSON

Operation

Theorem check of the sequence's own rule

{
  "operation": "theorem_check",
  "theorem": "liu_wang_three_term"
}

Claims

{
  "claim": "real_rooted",
  "from_index": 2,
  "region": {
    "upper": {
      "closed": true,
      "value": "0"
    }
  }
}

{
  "claim": "interlacing",
  "from_index": 3,
  "kind": "weak",
  "orientation": "previous_interlaces_next"
}

Certificate

theorem_check, SHA-256 7f502ae6d5f6db8e6c58a1c093418ca60e0c5f65f789227006c28b8475c95305, 5,544 bytes (1,606 compressed); download

polytool replayed the certificate when the step was written.

Cited by

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003