Proof step e-00224510dcd0cfb5

preserver_image hadamard_multiplier, shift 0 → concludes about Square array read by antidiagonals: S(p,q) = (p+q+1)!(2p+2q+1)!/((p+1)!(2p+1)!(q+1)!(2q+1)!) (p,q>=0). 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\":2,\"coefficient_offset\":1}]}}"
}

Premises

• One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318. (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 36928ba8d5e14fa1dd3ea9da76a2b1ae420c2ebcf5040559e84a344f649be731, 22,605 bytes (4,875 compressed); download JSON

Cited by

• n-af1d3c6f634a3212 (real_rooted)
• n-af1d3c6f634a3212 (one_signed_coefficients)
• n-af1d3c6f634a3212 (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.