Proof step e-b4af3e745574d56e

basis_map A008277 → concludes about basis_map_A008277 of Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n. JSON

Theorem

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?

Operation

Image under a basis map u^k -> P_k

{
  "map": "A008277",
  "operation": "basis_map"
}

Premises

• Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n. (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: basis_map, SHA-256 014a4ed0e5b002c5e4f6b6e39f65886cfd984588d2a5b1344f9474e082e623e1, 7,713 bytes (1,525 compressed); download JSON

Cited by

• n-f2ca92e21d86d85e (real_rooted)
• n-f2ca92e21d86d85e (one_signed_coefficients)
• n-f2ca92e21d86d85e (hadamard_kernel)

References

1. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741