Proof step e-92fa46b1f7f7427e

gamma_polynomial scale 1, shift 0 → concludes about gamma_polynomial of Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)*(2*n+3)/6)*T(n-2, k-1), 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) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)*(2*n+3)/6)*T(n-2, k-1), 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 be4db41ccfbc30284aa0c4ea511f288a984d6f75393eaaf5162a4b0426878609, 10,924 bytes (2,492 compressed); download JSON

Cited by

• n-95a6ecb670f8a672 (real_rooted)
• n-95a6ecb670f8a672 (one_signed_coefficients)
• n-95a6ecb670f8a672 (hadamard_kernel)