Proof step e-c483eeee9c2638c9

theorem_check liu_wang → concludes about basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1). shifted by x -> … JSON

Theorem

Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

Operation

Theorem check of the node's own rule

{
  "operation": "theorem_check",
  "theorem": "liu_wang"
}

Premises

None: this step is a direct theorem check.

Claims

• { "claim": "real_rooted", "from_index": 0, "region": { "upper": { "closed": false, "value": "0" } } }
• { "claim": "interlacing", "from_index": 1, "kind": "weak", "orientation": "previous_interlaces_next" }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: theorem_check, SHA-256 e66fb126b32dd90d30b811c775874c894650d257bd0274a6ce9dfc2da26cd306, 4,233 bytes (1,276 compressed); download JSON

Cited by

• n-0e6d847331141ca6 (real_rooted)
• n-0e6d847331141ca6 (one_signed_coefficients)
• n-0e6d847331141ca6 (hadamard_kernel)
• n-0e6d847331141ca6 (interlacing)

References

1. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003