Proof step e-a039cba7823b0b25

gamma_polynomial scale 1, shift 0 → concludes about Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one version of) Fibonacci polynomials. 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

• delannoy (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 d44d40049b95fd6a63c9afcfa609e9d7646a85f365045601067e1fdfc17c39a4, 10,467 bytes (2,373 compressed); download JSON

Cited by

• n-75ccd0453a9b6cec (real_rooted)
• n-75ccd0453a9b6cec (one_signed_coefficients)
• n-75ccd0453a9b6cec (hadamard_kernel)