Proof step e-ea229ed6e1820773

gamma_polynomial scale 1/2, shift 1 → concludes about Triangle read by rows: coefficients d(n,k) of André polynomials D(x,n) = Sum_{k>0} d(n,k)*x^k. JSON

Theorem

Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

Operation

Gamma polynomial of a palindromic parent

{
  "index_shift": 1,
  "operation": "gamma_polynomial",
  "scale": "1/2"
}

Premises

• eulerian (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": 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 9dc79543b4942f363e10e4906e85cbc790f7844efa3386415cef95bf08ad7ca1, 6,621 bytes (1,890 compressed); download JSON

Cited by

• n-b4814297dd192d81 (real_rooted)
• n-b4814297dd192d81 (one_signed_coefficients)
• n-b4814297dd192d81 (hadamard_kernel)