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, k) for n < 2 and binomial(n, k) + 4^(n-1) * binomial(n-2, k-1) otherwise.
Premises: none: a direct theorem check
Theorem: [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ?
Lean: RealRooted route sign RealRooted.threeTerm_interlaces_of_eval_nonpos, RealRooted.threeTerm_interlaces, RealRooted.eval_nonpos_of_natDegree_le_two
Checked: polytool replayed · Lean not attempted

Technical details

Step id

e-75cf58c1094a3b58 · 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"
    }
  }
}

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

Certificate

theorem_check, SHA-256 d5c5de3c4ca454161ac57c85dd387d13f1362500a0a0b91398b36e9aa06641b5, 5,682 bytes (1,665 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