Proof step e-9754e3fde53fc70a

gamma_polynomial scale 1, shift 0 → concludes about reversal of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by… 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 T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 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 4943ad95463ff262c3e24edc7de2207a6dba5a043fa97deaf669df3f18afdf88, 5,400 bytes (1,590 compressed); download JSON

Cited by

• n-5a3397c669570bd5 (real_rooted)
• n-5a3397c669570bd5 (one_signed_coefficients)
• n-5a3397c669570bd5 (hadamard_kernel)