Proof step

Row-hypergeometric rows (multiplier sequences)

Theorem check of the sequence's own rule hypergeometric row

Concludes: Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).
Premises: none: a direct theorem check
Theorem: [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ?
Lean: RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos, RealRooted.isPFMultiplierSequence_inv_ascPochhammer, RealRooted.IsPFMultiplierSequence.mul, RealRooted.IsPFMultiplierSequence.diagonalOperator_isPF
Checked: polytool replayed · Lean not attempted

Technical details

Step id

e-0e7bbe10cdbc75ef · JSON

Operation

Theorem check of the sequence's own rule

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

Claims

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

Certificate

theorem_check, SHA-256 8373e71f6a77d2fe8bd99e061c22d46d45dc15472d4c8ecbf812fd80cb846511, 4,831 bytes (1,769 compressed); download

polytool replayed the certificate when the step was written.

Cited by

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.