Proof step e-e49ea542bfed6bbf

gamma_polynomial scale 1, shift 1 → concludes about Triangle T(n,k) (n >= 1, 0 <= k <= floor((n-1)/2)) read by rows, where T(n,k) = (k+1)T(n-1,k) + (2n-4k)T(n-1,k-1). 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"
}

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 0baadac64d53ecdc879d2afc8184e3ce36704616bd5a3912abd5c6b727e017de, 6,609 bytes (1,890 compressed); download JSON

Cited by

• n-991b9bffcf8ddedb (real_rooted)
• n-991b9bffcf8ddedb (one_signed_coefficients)
• n-991b9bffcf8ddedb (hadamard_kernel)