Proof step e-5f21311bb67f8c1c

theorem_check liu_wang → concludes about Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the un… 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 a0ec723de98c4f166dadb3dcf8b252126ac491e89e8ba5e1c34fe8e99864a5b4, 2,735 bytes (1,031 compressed); download JSON

Cited by

• n-efe23a76953cf69d (real_rooted)
• n-efe23a76953cf69d (one_signed_coefficients)
• n-efe23a76953cf69d (hadamard_kernel)
• n-efe23a76953cf69d (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