Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver reversal, shift 0

Concludes

shift_x_plus_one of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).

Premises

basis_map_A013611 of Triangle of coefficients in expansion of (1+3*x)^n.
sequence: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]

Theorem

polytool ?

Lean

RealRooted route reversed RealRooted.X_pow_mul_reverse_eq_zero_or_splits, RealRooted.reflect_eq_zero_or_splits

Checked

polytool replayed · Lean not attempted

Technical details

Step id

e-04aceafc1533a6ef · JSON

Operation

Image under a real-rootedness preserver

{
  "index_shift": 0,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"
}

Claims

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

Certificate

preserver_image, SHA-256 91c78f0ca8ec38a4ff01178c362fa67e9ef62593cf3f1d740617014dc75c957a, 7,927 bytes (2,314 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-fcd6433895993258 (one_signed_coefficients)
n-fcd6433895993258 (hadamard_kernel)
n-fcd6433895993258 (real_rooted)