Proof step e-d07da0cb5a6ac36b

theorem_check liu_wang → concludes about Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows. 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 1ee9a85fa832b27b252e53ab2d756de256c666dbfee65c2e4da51ff79b853bcd, 2,767 bytes (1,031 compressed); download JSON

Cited by

• n-75a6475b3e851dc5 (real_rooted)
• n-75a6475b3e851dc5 (one_signed_coefficients)
• n-75a6475b3e851dc5 (hadamard_kernel)
• n-75a6475b3e851dc5 (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