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) = 2^k * (n!/k!)*binomial(n-1,k-1).
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-d29ec21b4af10167 · 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 813d4b443eb2b33a33160c45b135c8ee7c5989cf2bc2f68c9871a8b81abbfdb5, 5,542 bytes (1,855 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.