Proof step e-4d3032ecd7423900

preserver_image hadamard_multiplier, shift -1 → concludes about T(n,k) = binomial(n,2*k)*binomial(2*k,k) for 0 <= k <= n, triangle read by rows. JSON

Theorem

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

Lean (RealRooted): RealRooted.isMultiplierSequence_inv_factorial

Operation

Image under a real-rootedness preserver

{
  "index_shift": -1,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"hadamard_multiplier\",\"factor\":{\"terms\":[{\"kind\":\"inverse_factorial\"}]}}"
}

Premises

• Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polynomials with decreasi… (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": 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 56c8f23718ba9a590f3d23913e738d125a74cfe172c0ead280f96e49ae3a2868, 9,505 bytes (2,470 compressed); download JSON

Cited by

• n-4b168caf0bc63f8b (real_rooted)
• n-4b168caf0bc63f8b (one_signed_coefficients)
• n-4b168caf0bc63f8b (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.