Proof step e-c8375d3b3fe52ed3

preserver_image reversal, shift 0 → concludes about 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. JSON

Theorem

Affine, reversal and Möbius substitutions (elementary) polytool ?

Lean (RealRooted): route reversed RealRooted.X_pow_mul_reverse_eq_zero_or_splits, RealRooted.reflect_eq_zero_or_splits

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}}"
}

Premises

• Triangle related to the o.g.f.s. of the right-hand columns of A130534 (E(x,m=1,n)). (sequence)
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]

Claims

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

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: preserver_image, SHA-256 21afc54ec177d435ebb15edb0252b171b3d7387c95d264b7c160d69f1274b8cf, 14,008 bytes (2,931 compressed); download JSON

Cited by

• n-2eda446fe9bdb6ec (real_rooted)
• n-2eda446fe9bdb6ec (one_signed_coefficients)
• n-2eda446fe9bdb6ec (hadamard_kernel)