Proof step e-cc6de87a980d771e

gamma_polynomial scale 1, shift 0 → concludes about gamma_polynomial of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1). 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, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1). (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 0ca3255e6503550df5d6222aede9fc23c8b4c11c6668072b0dafbdffef27aa98, 10,993 bytes (2,312 compressed); download JSON

Cited by

• n-a8a898e8495cff05 (real_rooted)
• n-a8a898e8495cff05 (one_signed_coefficients)
• n-a8a898e8495cff05 (hadamard_kernel)