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_A013610 of Triangle of coefficients in expansion of (1+2*x)^n.

Premises

basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.
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-c8ba40b4a1835d23 · 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 d5dacf61ec71fe50ca0b93fe936375b4737f3dba5ef92084b39e19e304f73b44, 7,923 bytes (2,321 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-5c2409b346bb76f2 (one_signed_coefficients)
n-5c2409b346bb76f2 (hadamard_kernel)
n-5c2409b346bb76f2 (real_rooted)