Proof step e-7fb7a0f59a4bd766

gamma_polynomial scale 1, shift 0 → concludes about gamma_polynomial of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 5. 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) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 5. (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 c2b364f4dbf2d66ce7e9f0e2e8822dfe2eb78a49342a171c49cdf3d4389654a2, 10,204 bytes (2,346 compressed); download JSON

Cited by

• n-ca46250f05be6e7c (real_rooted)
• n-ca46250f05be6e7c (one_signed_coefficients)
• n-ca46250f05be6e7c (hadamard_kernel)