Proof step e-55e5f4941dd58e15

gamma_polynomial scale 1, shift 0 → concludes about Triangular array of Motzkin polynomial coefficients. 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 of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan triangle. (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 10a546576defe57674614aa5eb7de31663212bac12ad3fed0e031a46f1f39528, 10,641 bytes (2,385 compressed); download JSON

Cited by

• n-169bd483754ab390 (real_rooted)
• n-169bd483754ab390 (one_signed_coefficients)
• n-169bd483754ab390 (hadamard_kernel)