Proof step e-db4d4459bbf9061f

basis_map A008277 → concludes about basis_map_A008277 of Triangle: row #n has n+1 terms. T(n,m) = 4^m (2n+1)! / ( (2n-2m)! (2m+1)! ). 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: row #n has n+1 terms. T(n,m) = 4^m (2n+1)! / ( (2n-2m)! (2m+1)! ). (sequence)
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]

Claims

• { "claim": "real_rooted", "from_index": 0, "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 6e9222794301553bfcc9ae661a14db2e4999d9307bf6841b3153d40b91df06aa, 12,669 bytes (2,268 compressed); download JSON

Cited by

• n-7f78ef09e94071ba (real_rooted)
• n-7f78ef09e94071ba (one_signed_coefficients)
• n-7f78ef09e94071ba (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