Proof step

Ray basis map theorem

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

Concludes

Premises

Triangle read by rows: T(n,k) = (n+1)*(n+2)*(k+3)*binomial(n,k)/6, 0 <= k <= n.
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-e51fcf90a18be50b · 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 7959ebab0f188c8144e32fb8e2a0871105ad4cdb8de03e777c88805b335c9400, 14,065 bytes (2,584 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