Proof step

Ray basis map theorem

Image under a basis map u^k -> P_k A008277

Concludes

basis_map_A008277 of (3,1) Pascal triangle.

Premises

(3,1) Pascal triangle.
sequence: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]

Theorem

Checked

polytool replayed · Lean not attempted

Technical details

Step id

e-de3b8dddb87b1aad · JSON

Operation

Image under a basis map u^k -> P_k

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

Claims

{
  "claim": "real_rooted",
  "from_index": 0,
  "region": {
    "upper": {
      "closed": true,
      "value": "0"
    }
  }
}

Certificate

basis_map, SHA-256 f3d730da3e0abcf0cc61320d0a8beb02a2f7fd3cd2ccc63e4575236121268517, 9,479 bytes (2,001 compressed); download

polytool replayed the certificate when the step was written.

Cited by

References

[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