Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver reversal, shift 0

Concludes

shift_x_plus_one of E2(n, k), the second-order Eulerian numbers with E2(0, k) = δ_{0, k}. Triangle read by rows, E2(n, k) for 0 <= k <= n.

Premises

Coefficient triangle for polynomials used for e.g.f.s for unsigned Stirling1 diagonals.
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-c735f431717d17a0 · 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 abbb76f3e6f212f0414fb767ec1e24fd78f72988a48b3b9d8806678cfa73fc10, 15,845 bytes (3,077 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-95a09e5daac73697 (one_signed_coefficients)
n-95a09e5daac73697 (hadamard_kernel)
n-95a09e5daac73697 (real_rooted)