Proof step e-418ae03de1c699ba

gamma_polynomial scale 1, shift 0 → concludes about gamma_polynomial of Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(x) = x + 3. 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) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(x) = x + 3. (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": 1, "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 c8d0b5b0e721b42196cebb53324838fe7f86827e537d1bd361ab8badfb0a5f50, 10,217 bytes (2,353 compressed); download JSON

Cited by

• n-9fdebfaf3bddb35b (real_rooted)
• n-9fdebfaf3bddb35b (one_signed_coefficients)
• n-9fdebfaf3bddb35b (hadamard_kernel)