Proof step e-4c4f74d9ba733cff

gamma_polynomial scale 1, shift 0 → concludes about Triangle of coefficients in expansion of (1+2*x)^n. 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) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n. (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 26c7c2bc659eb913df6fa559691ea5fa84713a59e1fdc4c8aa3043e5511fb0a0, 5,427 bytes (1,612 compressed); download JSON

Cited by

• n-71965e79e47a4526 (one_signed_coefficients)
• n-71965e79e47a4526 (hadamard_kernel)
• n-71965e79e47a4526 (real_rooted)