Proof step e-419a6747cb084155

theorem_check liu_wang → concludes about Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned… 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 addfe6bafcc78af13d1308a4b02f1b8c882d8713f898529a6cf371ea40c5a225, 2,735 bytes (1,033 compressed); download JSON

Cited by

• n-4d60c3dd8627c01d (real_rooted)
• n-4d60c3dd8627c01d (one_signed_coefficients)
• n-4d60c3dd8627c01d (hadamard_kernel)
• n-4d60c3dd8627c01d (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