Proof step e-41c16cba2e1a2f2b

gamma_polynomial scale 1, shift 0 → concludes about reversal of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0 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

• basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0 (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 d34130e610affc495af412e976c0975fd5b73f76973ea64b9b8f13e49d1145e3, 5,361 bytes (1,560 compressed); download JSON

Cited by

• n-67e8e5fe38a23ea9 (real_rooted)
• n-67e8e5fe38a23ea9 (one_signed_coefficients)
• n-67e8e5fe38a23ea9 (hadamard_kernel)