Proof step e-98288ca18963a535

basis_map A008277 → concludes about basis_map_A008277 of Triangle read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry equal to k (n >= 0; 0 <= k <= n/2). 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 read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry equal to k (n >= 0; 0 <= k <= n/2). (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 b6e1ffb0fb34e2e31b5f68f8d428726371dea28c776cde32767aa9aa6c3a82a3, 7,555 bytes (1,512 compressed); download JSON

Cited by

• n-0c1c15b87c16cb0d (real_rooted)
• n-0c1c15b87c16cb0d (one_signed_coefficients)
• n-0c1c15b87c16cb0d (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