Proof step e-e2e7a7b682b47947

gamma_polynomial scale 1, shift 0 → concludes about Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C(n+1,2*k+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

• One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318. (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 40c04eed59dd1c97fae4c6570133e875e3c14ab85c5a74e89dcbe8970e7a1a4a, 10,220 bytes (2,306 compressed); download JSON

Cited by

• n-cf38558d99f4fe37 (real_rooted)
• n-cf38558d99f4fe37 (one_signed_coefficients)
• n-cf38558d99f4fe37 (hadamard_kernel)