Proof step e-b8c83da274fc75bd

gamma_polynomial scale 1, shift 0 → concludes about gamma_polynomial of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j. 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 whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j. (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 fca9fc4b2a409f9c708f50fda8bd67ab106311d4fb0cd974735de8a34a097acd, 5,340 bytes (1,559 compressed); download JSON

Cited by

• n-d09dd8597f391f42 (one_signed_coefficients)
• n-d09dd8597f391f42 (hadamard_kernel)
• n-d09dd8597f391f42 (real_rooted)