Proof step e-aaeb3397827b8bbd

preserver_image hadamard_multiplier, shift 2 → concludes about Triangular array: T(n,k) = sqrt(C(n-1,k-1)*C(n-1,k)*C(n,k+1)* C(n+1,k+1)*C(n+1,k)*C(n,k-1)), where C(n,k) = binomial(n,k). JSON

Theorem

Pólya–Schur theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?

Operation

Image under a real-rootedness preserver

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

Premises

• Triangle of Dyck paths counted by number of long interior inclines. (sequence)
  uses: Every row P_n, n >= 3, is zero or real-rooted with all roots in (-inf, 0]

Claims

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

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: preserver_image, SHA-256 ae7baffb4fb7f45b3e62a1927fda890eb1941f01579fafb7f0af89ec9600f404, 28,056 bytes (6,506 compressed); download JSON

Cited by

• n-05b91207a1ba3acf (real_rooted)
• n-05b91207a1ba3acf (one_signed_coefficients)
• n-05b91207a1ba3acf (hadamard_kernel)

References

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