Proof step e-a23b1eece2788390

theorem_check liu_wang → concludes about Triangle read by rows, coefficients of the polynomials P(k, x) = (1/2) Sum_{p=0..k-1} Stirling2(k, p+1)*x^p*(1-4*x)^(k-1-p)*(2*p+2)!/(p+1)!. 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 9e8a7de88687ac3f60becd0bf970f88dc24ed6e67244e0e06877f124e6008d81, 3,297 bytes (1,136 compressed); download JSON

Cited by

• n-633aaaf6fe98dc51 (real_rooted)
• n-633aaaf6fe98dc51 (one_signed_coefficients)
• n-633aaaf6fe98dc51 (hadamard_kernel)
• n-633aaaf6fe98dc51 (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