Proof step e-c990b380b99b5cea

theorem_check liu_wang → concludes about Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 6*n + 3. 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 06f36c31a6e59836f9aecfd5d42e6ce02035479dcfbe6061ec7588ca91b72621, 3,299 bytes (1,133 compressed); download JSON

Cited by

• n-444025a5f2f6e0e5 (real_rooted)
• n-444025a5f2f6e0e5 (one_signed_coefficients)
• n-444025a5f2f6e0e5 (hadamard_kernel)
• n-444025a5f2f6e0e5 (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