Proof step e-d5c3213d2b3c5934

basis_map A008277 → concludes about basis_map_A008277 of Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); .... 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 read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); .... (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 900d1e15dfeab9e5d921e36bf64f8fac69dbd6287a73f4a5da8dfeaa3378e21e, 8,934 bytes (1,949 compressed); download JSON

Cited by

• n-01c68de89070fe18 (real_rooted)
• n-01c68de89070fe18 (one_signed_coefficients)
• n-01c68de89070fe18 (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