Proof step e-13742d707ef3ea03

basis_map A008277 → concludes about basis_map_A008277 of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1). 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) = binomial(n+k-1, 2*k-1). (sequence)
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]

Claims

• { "claim": "real_rooted", "from_index": 1, "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 f61c0ab010e2aeb263067079cec7f61e9c75479b32b4b0bd77f2bec4dc3f3d30, 12,029 bytes (2,163 compressed); download JSON

Cited by

• n-e739b38b5baf2db5 (real_rooted)
• n-e739b38b5baf2db5 (one_signed_coefficients)
• n-e739b38b5baf2db5 (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