Proof step e-d8c50adb346fbffe

gamma_polynomial scale 1, shift 0 → concludes about gamma_polynomial of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k). 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 T(n, k) read by rows, T(n, k) = n!*binomial(n, k). (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 b09377dc37cd9de640eeb508e83667f2ff2e0fec53c4e9cadc6a87ad6bd5a176, 5,904 bytes (1,774 compressed); download JSON

Cited by

• n-665a2c7873826b72 (real_rooted)
• n-665a2c7873826b72 (one_signed_coefficients)
• n-665a2c7873826b72 (hadamard_kernel)