Proof step e-e37e9c5997239324

preserver_image hadamard_multiplier, shift 2 → concludes about Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m). 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\":1,\"coefficient_offset\":0}]}}"
}

Premises

• Triangle read by rows: T(n, k) = binomial(2n, 2k). (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 d730327b585ad78c0f10a988767d2f30b08f06bf5831c2e72dd69e0744339684, 25,846 bytes (5,881 compressed); download JSON

Cited by

• n-b002679da3d71a0f (real_rooted)
• n-b002679da3d71a0f (one_signed_coefficients)
• n-b002679da3d71a0f (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.