Proof step e-fbbe2b618491014f

gamma_polynomial scale 1, shift 0 → concludes about gamma_polynomial of Generalized Narayana numbers, T(n, k) = 3/(n + 1)*binomial(n + 1, k + 2)*binomial(n + 1, k - 1), triangular array 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

• Generalized Narayana numbers, T(n, k) = 3/(n + 1)*binomial(n + 1, k + 2)*binomial(n + 1, k - 1), triangular array read by rows. (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 6c78d1613da4d1e62bff7ed72a46208d1412b4554ccf92ac0ca6c6c307574bb8, 11,017 bytes (2,492 compressed); download JSON

Cited by

• n-92efd0b8c392ac45 (real_rooted)
• n-92efd0b8c392ac45 (one_signed_coefficients)
• n-92efd0b8c392ac45 (hadamard_kernel)