Proof step

Laguerre’s theorem on differential operators

Image under a real-rootedness preserver derivative polynomial, shift 0

Concludes

Triangle read by rows: A008277 * A007318.

Premises

Triangle of numbers a(n,k), 0 <= k <= n: number of set partitions of {1,2,...,n} in which exactly k of the blocks have been distinguished.
sequence: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]

Theorem

[SP14] polytool ?

Lean

RealRooted RealRooted.splits_add_C_mul_derivative_all, RealRooted.splits_C_mul_add_derivative, Polynomial.Splits.derivative

Checked

polytool replayed · Lean not attempted

Technical details

Step id

e-6ac9b6d92b30c23f · JSON

Operation

Image under a real-rootedness preserver

{
  "index_shift": 0,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"
}

Claims

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

Certificate

preserver_image, SHA-256 45a38429de4cd7817bcc530eaf181c0b7cde516727a2dcd5cb3d58bb0094ce13, 8,565 bytes (2,387 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-bdae5aa3d1cac0ec (one_signed_coefficients)
n-bdae5aa3d1cac0ec (hadamard_kernel)
n-bdae5aa3d1cac0ec (real_rooted)

References

[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.