Proof step
Ray basis map theorem
Image under a basis map u^k -> P_k A008277
- Concludes
- basis_map_A008277 of theta_plus_one of a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
- Premises
- derivative of basis-map coefficients of A scaled Hermite triangle.
sequence: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
- derivative of basis-map coefficients of A scaled Hermite triangle.
- Theorem
- [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?
- Checked
- polytool replayed · Lean not attempted
Technical details
- Step id
e-7321dc14a1ffd2df· JSON- Operation
- Image under a basis map u^k -> P_k
{ "map": "A008277", "operation": "basis_map" } - Claims
{ "claim": "real_rooted", "from_index": 1, "region": { "upper": { "closed": true, "value": "0" } } }- Certificate
basis_map, SHA-25688e986b1d833113083e1ced27e4431898026e4c1724573446fb0dbda60db16af, 13,472 bytes (2,456 compressed); downloadpolytool replayed the certificate when the step was written.- Cited by
n-944d25c7efd08f90(real_rooted)n-944d25c7efd08f90(one_signed_coefficients)n-944d25c7efd08f90(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