Proof step e-6239f8c0a2c99666

gamma_polynomial scale 1, shift 0 → concludes about reversal of Triangle read by rows: row n consists of n zeros followed by n+1. 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 of denominators in Leibniz's Harmonic Triangle a(n,k), n >= 1, 1 <= k <= n. (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": 1, "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 46aa3c60bd1bc8b41e98a15de1ed2460592573aaebc666165f725738a2adc2fa, 11,216 bytes (2,622 compressed); download JSON

Cited by

• n-dabe0dfba915bf52 (real_rooted)
• n-dabe0dfba915bf52 (one_signed_coefficients)
• n-dabe0dfba915bf52 (hadamard_kernel)