Proof step

Pólya–Schur theorem on multiplier sequences

Image under a real-rootedness preserver hadamard multiplier, shift 2

Concludes

Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m).

Premises

Triangle read by rows: T(n, k) = binomial(2n, 2k).
sequence: Every row P_n, n >= 3, is zero or real-rooted with all roots in (-inf, 0]

Theorem

[SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ?

Checked

polytool replayed · Lean not attempted

Technical details

Step id

e-dbc1a9d96bc2026c · JSON

Operation

Image under a real-rootedness preserver

{
  "index_shift": 2,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"hadamard_multiplier\",\"factor\":{\"terms\":[{\"kind\":\"binomial\",\"index_offset\":1,\"coefficient_offset\":0}]}}"
}

Claims

{
  "claim": "real_rooted",
  "from_index": 1,
  "region": {
    "upper": {
      "closed": true,
      "value": "0"
    }
  }
}

Certificate

preserver_image, SHA-256 70311a273ef4b522a1fbab791a8c39a56a48b68813972985255460b2e5a5a8d6, 25,981 bytes (5,919 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-b002679da3d71a0f (real_rooted)
n-b002679da3d71a0f (one_signed_coefficients)
n-b002679da3d71a0f (hadamard_kernel)

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.