Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver reversal, shift 0

Concludes

Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the first kind (n >= 2; 1 <= k <= n-1; s(n,j) =…

Premises

Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....
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-371061716b8e3d0a · 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 7ed1f86e4f51a6fe28a660dd3abce3577456f1ba250a5d86366523a7c4a00f01, 9,499 bytes (2,505 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-914700b1060feeb2 (one_signed_coefficients)
n-914700b1060feeb2 (hadamard_kernel)
n-914700b1060feeb2 (real_rooted)