Proof step
Ray basis map theorem
Image under a basis map u^k -> P_k A008277
- Concludes
- one_plus_derivative of basis_map_A008277 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.
- Premises
- basis_map_A007318 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_A007318 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-d0627c8203c99c93· 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-2562beaae3421416e14bb02d5cae4d2c9d1cfddd994669c5eeeb3aafe70bdc8c76f, 9,078 bytes (1,950 compressed); downloadpolytool replayed the certificate when the step was written.- Cited by
n-5bd42e161dd707c1(real_rooted)n-5bd42e161dd707c1(one_signed_coefficients)n-5bd42e161dd707c1(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