Proof step

Row-hypergeometric rows (multiplier sequences)

Theorem check of the sequence's own rule hypergeometric row

Concludes: Triangle read by rows. T(n, k) = binomial(n - 1, k - 1)*(n + k)! / k!.
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-5a2561bd80ba91ec · 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": true,
      "value": "0"
    }
  }
}

Certificate

theorem_check, SHA-256 05886e880fac8ab6b2ca9205525ed7342a54fcb584152273ec4791c54a9863b4, 6,323 bytes (2,029 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.