Proof step e-98e347ce1f066bf3

theorem_check liu_wang → concludes about Triangle T(n,k) read by rows: T(n,k) = 1 if k=1 or k=n, otherwise T(n,k) = (3*n-3*k+1)*T(n-1,k-1) + (3*k-2)*T(n-1,k). 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 b200918bd4408524b247945d17923ddc73a297d063987321d3d76547b09aa0be, 3,299 bytes (1,121 compressed); download JSON

Cited by

• n-ddb0cd3a4d6edf09 (real_rooted)
• n-ddb0cd3a4d6edf09 (one_signed_coefficients)
• n-ddb0cd3a4d6edf09 (hadamard_kernel)
• n-ddb0cd3a4d6edf09 (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