Proof step e-510754e51c3ada00

gamma_polynomial scale 1, shift 0 → concludes about gamma_polynomial of Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps and contain exactly k steps to the east, p… JSON

Theorem

Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

Operation

Gamma polynomial of a palindromic parent

{
  "index_shift": 0,
  "operation": "gamma_polynomial",
  "scale": "1"
}

Premises

• Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps and contain exactly k steps to the east, p… (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: gamma_polynomial, SHA-256 07559943f7d4f5ba6826b5cc7e4707ddf276ed1f838f67f6ff3775d3a958db5f, 11,050 bytes (2,493 compressed); download JSON

Cited by

• n-ce7f61ca1eb91390 (real_rooted)
• n-ce7f61ca1eb91390 (one_signed_coefficients)
• n-ce7f61ca1eb91390 (hadamard_kernel)