Proof step
Ray basis map theorem
Image under a basis map u^k -> P_k A008277
- Concludes
- basis_map_A008277 of Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 2^(n-1) * binomial(n-2, k-1) otherwise.
- Premises
- Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 2^(n-1) * binomial(n-2, k-1) otherwise.
sequence: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
- Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 2^(n-1) * binomial(n-2, k-1) otherwise.
- Theorem
- [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?
- Checked
- polytool replayed · Lean not attempted
Technical details
- Step id
e-37984d7925024601· 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-256abc2855ed76ecfba20f9c0edddb38e378f8e9f0c9fe89cd8ad55988698c305b3, 15,487 bytes (2,923 compressed); downloadpolytool replayed the certificate when the step was written.- Cited by
n-fc3066c48ace6df3(one_signed_coefficients)n-fc3066c48ace6df3(hadamard_kernel)n-fc3066c48ace6df3(real_rooted)
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