Proof step e-94f70be4e24ec504

gamma_polynomial scale 1, shift 0 → concludes about gamma_polynomial of Triangle read by rows: T(n, k) = binomial(2n, 2k). 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) = binomial(2n, 2k). (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 64d39750dd9e579f5d836ea0f300decd0c535c12851bfd1cc8e643840472cbe4, 10,186 bytes (2,287 compressed); download JSON

Cited by

• n-4399296e302f2e9b (real_rooted)
• n-4399296e302f2e9b (one_signed_coefficients)
• n-4399296e302f2e9b (hadamard_kernel)