Proof step e-d938b07605a9ed19

gamma_polynomial scale 1, shift 0 → concludes about gamma_polynomial of Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^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 T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2,… (sequence)
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]

Claims

• { "claim": "real_rooted", "from_index": 1, "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 bcd5c651cd45c90a8ac7b5ca4bf106cb35a4827ac2a2ed6bf9c7ce9a1840bd4d, 6,861 bytes (1,999 compressed); download JSON

Cited by

• n-17caa2f11eb7e9b8 (real_rooted)
• n-17caa2f11eb7e9b8 (one_signed_coefficients)
• n-17caa2f11eb7e9b8 (hadamard_kernel)