Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver mobius, shift 0

Concludes

basis_map_A013609 of Triangle of coefficients in expansion of (1+3*x)^n.

Premises

basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.
sequence: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]

Theorem

polytool ?

Lean

RealRooted route fTransform RealRooted.splits_fPolynomial

Checked

polytool replayed · Lean not attempted

Technical details

Step id

e-43006129e543f8a9 · JSON

Operation

Image under a real-rootedness preserver

{
  "index_shift": 0,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"
}

Claims

{
  "claim": "real_rooted",
  "from_index": 0,
  "region": {
    "lower": {
      "closed": true,
      "value": "-1"
    },
    "upper": {
      "closed": true,
      "value": "0"
    }
  }
}

Certificate

preserver_image, SHA-256 072ca445cf19fd43f6ea5d53d94d0b055938343cbac7d0681f217a8c7c9a6648, 8,183 bytes (2,356 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-4f34a2344b135a41 (one_signed_coefficients)
n-4f34a2344b135a41 (hadamard_kernel)
n-4f34a2344b135a41 (real_rooted)