Proof step e-835093125a0f64b6

preserver_image hadamard_multiplier, shift 0 → concludes about Triangle of cubed binomial coefficients, T(n,k) = C(n,k)^3, read by rows. JSON

Theorem

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

Operation

Image under a real-rootedness preserver

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

Premises

• Square the entries of Pascal's triangle. (sequence)
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]

Claims

• { "claim": "real_rooted", "from_index": 0, "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 45519ff56043690e7de04bf293b9b8ba042021c1604d7be50d905e469b604c4b, 21,379 bytes (4,275 compressed); download JSON

Cited by

• n-fabb0a68e001a863 (real_rooted)
• n-fabb0a68e001a863 (one_signed_coefficients)
• n-fabb0a68e001a863 (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.