Proof step e-c52960bc5e97caa0

basis_map A008277 → concludes about Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n. 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

• basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. (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 584a3915a83173da0f2b8ec160dfb224b58b2f200df47e7f851db6e373d82fee, 7,525 bytes (1,487 compressed); download JSON

Cited by

• n-3cb22164e2f83fe1 (one_signed_coefficients)
• n-3cb22164e2f83fe1 (hadamard_kernel)
• n-3cb22164e2f83fe1 (real_rooted)

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