Proof step e-18b4793e32bd250f

basis_map A154602 → concludes about Exponential Riordan array [exp(sinh(x)*exp(x)), sinh(x)*exp(x)]. 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": "A154602",
  "operation": "basis_map"
}

Premises

• basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. (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": "-1" } } }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: basis_map, SHA-256 7be2569ce793b7954765a6d17bbcbc74aadd6c170a5f9721f9d37028c18e6067, 7,541 bytes (1,514 compressed); download JSON

Cited by

• n-806f02861480e04e (real_rooted)
• n-806f02861480e04e (one_signed_coefficients)
• n-806f02861480e04e (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