Proof step e-857a86cbb6c7881e

basis_map A008277 → concludes about Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0 <= 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 Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0 <= k <= n. (sequence)
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 1]

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 5dbea33f5f1b028325c49d7d9ae2cd84bbfd7f9c5bc20d4f0e3680abe026117a, 7,528 bytes (1,500 compressed); download JSON

Cited by

• n-e65f982fa1b436c5 (real_rooted)
• n-e65f982fa1b436c5 (one_signed_coefficients)
• n-e65f982fa1b436c5 (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