Proof step e-00ceaf3777abe095

gamma_polynomial scale 1/4, shift 1 → concludes about Triangle read by rows: T(n,k) (n >= 1, 0 <= k <= ceiling(n/2)-1) = number of permutations of [n] with k peaks. JSON

Theorem

Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

Operation

Gamma polynomial of a palindromic parent

{
  "index_shift": 1,
  "operation": "gamma_polynomial",
  "scale": "1/4"
}

Premises

• eulerian (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": 0, "region": { "upper": { "closed": true, "value": "0" } } }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: gamma_polynomial, SHA-256 2b96c6aae415fe3c6e041772f7eb71fe49be1b6ebbd52b2c4e92dfa0ec98c02b, 7,266 bytes (2,082 compressed); download JSON

Cited by

• n-3dbfcbca0c0c8d13 (real_rooted)
• n-3dbfcbca0c0c8d13 (one_signed_coefficients)
• n-3dbfcbca0c0c8d13 (hadamard_kernel)