Proof step e-00d47151fcd2720e

preserver_image reversal, shift 0 → concludes about T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows. 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\":1,\"first_degree\":0,\"step\":1}}"
}

Premises

• Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral … (sequence)
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]

Claims

• { "claim": "real_rooted", "from_index": 1, "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 86a8a8d2fd5e94cfe906298f0344dbff1a624452ad431c8f61052d1e863969fa, 14,066 bytes (3,217 compressed); download JSON

Cited by

• n-4689fb1386d440ad (one_signed_coefficients)
• n-4689fb1386d440ad (hadamard_kernel)
• n-4689fb1386d440ad (real_rooted)