Proof step e-36556341d5271807

gamma_polynomial scale 1, shift 0 → concludes about gamma_polynomial of Triangle T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2), read by rows. 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) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2), read by rows. (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 05f3a9972995d2475bb2b0b31b0647fa7ca54f9e695c1071784a89791956818e, 10,741 bytes (2,458 compressed); download JSON

Cited by

• n-eea7ec082f5ce396 (real_rooted)
• n-eea7ec082f5ce396 (one_signed_coefficients)
• n-eea7ec082f5ce396 (hadamard_kernel)