Proof step e-b469c0e3f65675eb

gamma_polynomial scale 1, shift 0 → concludes about gamma_polynomial of Triangle read by rows: T(n, k) = (6*n-6*k+1)*T(n-1, k-1) + (6*k-5)*T(n-1, k) + 6*T(n-2, k-1) with T(n, 1) = T(n, n) = 1. 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 read by rows: T(n, k) = (6*n-6*k+1)*T(n-1, k-1) + (6*k-5)*T(n-1, k) + 6*T(n-2, k-1) with T(n, 1) = T(n, n) = 1. (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 a983cae0511583184753880c7b7e14547efeeda2a3bf8532e7b5d3d1400e80bc, 11,282 bytes (2,509 compressed); download JSON

Cited by

• n-a2a51c20d79421a9 (real_rooted)
• n-a2a51c20d79421a9 (one_signed_coefficients)
• n-a2a51c20d79421a9 (hadamard_kernel)