Proof step e-ac43a5abaccfbd29

basis_map A008277 → concludes about basis_map_A008277 of Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318. 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

• Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318. (sequence)
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]

Claims

• { "claim": "real_rooted", "from_index": 1, "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 b9a7f2c0ea209df23ad357921fbae8a8867a717510b7cfe9df8f38b14cf1bd78, 12,638 bytes (2,252 compressed); download JSON

Cited by

• n-be27703e6965ccb1 (real_rooted)
• n-be27703e6965ccb1 (one_signed_coefficients)
• n-be27703e6965ccb1 (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