Proof step e-a9c75dc8bc8e63f9

gamma_polynomial scale 1, shift 0 → concludes about gamma_polynomial of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n). 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) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n). (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 517b785e68d78d6cfe29932141d7fc9d4fb9f7f665f899c5dc889ee6bde8cc52, 5,958 bytes (1,814 compressed); download JSON

Cited by

• n-3890202342120839 (real_rooted)
• n-3890202342120839 (one_signed_coefficients)
• n-3890202342120839 (hadamard_kernel)