Proof step e-64b74724257dbeaa

gamma_polynomial scale 1, shift 0 → concludes about Triangle T(n,k) = T(n-1,k) + 3*T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows. 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

• Pascal-(1,3,1) array. (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 bf125c02e7e084a241e89da4187b69bc01508ff54a39c2c1956befd03dc1cd89, 10,482 bytes (2,390 compressed); download JSON

Cited by

• n-e189038e95d51606 (real_rooted)
• n-e189038e95d51606 (one_signed_coefficients)
• n-e189038e95d51606 (hadamard_kernel)