Proof step e-78c582ff5f7b6a70

gamma_polynomial scale 1, shift 0 → concludes about Triangle read by rows, A007318 rows repeated. 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

• Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = k <= n-1, T(n,k) = T… (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 c86e9e6efd9e227f5b43c4135dce454901dcce028bb161da1da2ef3c68de6060, 5,559 bytes (1,629 compressed); download JSON

Cited by

• n-4124f4242e7d4c67 (real_rooted)
• n-4124f4242e7d4c67 (one_signed_coefficients)
• n-4124f4242e7d4c67 (hadamard_kernel)