Proof step

Liu–Wang theorem for three-term recurrences

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

Concludes: Triangle read by rows: T(n,k) = binomial(n-2, k-1) + n*binomial(n-1, k-1), 1 <= k <= n, starting with T(1, 1) = 1.
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-b4e5a61cd5085168 · JSON

Operation

Theorem check of the sequence's own rule

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

Claims

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

Certificate

theorem_check, SHA-256 075ed997a434087ca2798ca226bbceefddeba3aca3840eaaa866bcca218a64d4, 5,402 bytes (1,623 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