Proof step
Ray basis map theorem
Image under a basis map u^k -> P_k A008277
- Concludes
- basis_map_A008277 of basis_map_A013609 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.
- Premises
- basis_map_A013609 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.
sequence: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
- basis_map_A013609 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.
- Theorem
- [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?
- Checked
- polytool replayed · Lean not attempted
Technical details
- Step id
e-a8d08807a31639f7· 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-25677e30fa2b48e0620f581be7916a0175e3863b8d1145f401cfe4e7e57285d0924, 9,082 bytes (1,959 compressed); downloadpolytool replayed the certificate when the step was written.- Cited by
n-0b75221cfe4c7721(real_rooted)n-0b75221cfe4c7721(one_signed_coefficients)n-0b75221cfe4c7721(hadamard_kernel)
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