Proof step e-4347e41634c98e26

gamma_polynomial scale 1, shift 0 → concludes about gamma_polynomial of Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2). 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, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2). (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 c018c900bccb188b3b3565e8944081eb6eadf981293f58e8eb1e91b53e4b7f0b, 9,856 bytes (2,259 compressed); download JSON

Cited by

• n-4980330517acdfb6 (real_rooted)
• n-4980330517acdfb6 (one_signed_coefficients)
• n-4980330517acdfb6 (hadamard_kernel)