Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver reversal, shift 0

Concludes

basis_map_A008277 of Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. …

Premises

Triangular array A066667 or A008297 unsigned and transposed.
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-d55eff0a3758efaf · 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 2c50a1017c28e6b2008ef5fbdbcfb7c875967ea4f7d822d0c2378141d13a9725, 10,637 bytes (2,698 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-011d67b73ad91983 (real_rooted)
n-011d67b73ad91983 (one_signed_coefficients)
n-011d67b73ad91983 (hadamard_kernel)