Proof step e-a112a311030a6f8e

gamma_polynomial scale 1, shift 0 → concludes about gamma_polynomial of Triangle read by rows: T(n,k) (1<=k<=n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), whe… 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) (1<=k<=n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), whe… (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 0f0dc47c7fde0dd9bfaa8f35e9032e866b453fc957cb7970f26c1aed4371ecf1, 10,259 bytes (2,365 compressed); download JSON

Cited by

• n-c258d63c1dbe15cd (real_rooted)
• n-c258d63c1dbe15cd (one_signed_coefficients)
• n-c258d63c1dbe15cd (hadamard_kernel)