Proof step e-bfe3673616f49c96

theorem_check liu_wang_derivative_lag → concludes about Triangle read by rows: T(n, k) = (6*n-6*k+1)*T(n-1, k-1) + (6*k-5)*T(n-1, k) + 6*T(n-2, k-1) with T(n, 1) = T(n, n) = 1. JSON

Theorem

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

Lean (RealRooted): route halfGrowth RealRooted.derivLag_strictInterl_of_nonnegCoeffs, RealRooted.derivLag_strictInterl_and_noCommonRoot, RealRooted.derivLag_eval_zero_pos

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 3c5c5cfb68189abd3e33b653fed43306ea186c6622f875c9f4be2e6b9fd1c6dc, 5,490 bytes (1,461 compressed); download JSON

Cited by

• n-a0dd3d6ef57da7a5 (real_rooted)
• n-a0dd3d6ef57da7a5 (one_signed_coefficients)
• n-a0dd3d6ef57da7a5 (hadamard_kernel)
• n-a0dd3d6ef57da7a5 (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