Proof step e-f727ccd9a8e875c3

theorem_check liu_wang_derivative_lag → concludes about Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)/2)*T(n-2, k-1), read by rows. JSON

Theorem

Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

Lean (RealRooted): route sign RealRooted.threeTerm_interlaces_of_nonnegCoeffs, RealRooted.LiuWang.strictInterl_nonpos_lag_sequence

Operation

Theorem check of the node's own rule

{
  "operation": "theorem_check",
  "theorem": "liu_wang_derivative_lag"
}

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": "strict", "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 2ef511d883e8cda855bf3376a3986b0306a240d3294bfec965bd6213605cf35a, 5,002 bytes (1,464 compressed); download JSON

Cited by

• n-8c29b5d8d22f0df0 (real_rooted)
• n-8c29b5d8d22f0df0 (one_signed_coefficients)
• n-8c29b5d8d22f0df0 (hadamard_kernel)
• n-8c29b5d8d22f0df0 (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