Proof step e-babfdb0b6fa40ea8

gamma_polynomial scale 1, shift 0 → concludes about gamma_polynomial of Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, ... 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) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, ... (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 ecc21755662a87e83be3046c52c9b1f08fcc7e6cfca028583c95651bf6713409, 10,627 bytes (2,451 compressed); download JSON

Cited by

• n-a5662819bbc56ccb (real_rooted)
• n-a5662819bbc56ccb (one_signed_coefficients)
• n-a5662819bbc56ccb (hadamard_kernel)