Proof step e-b9d2d25190d8d499

theorem_check liu_wang → concludes about Triangle read by rows: T(n, k) = 2^n * [x^k] hypergeom([-n, -n, 1], [1], x). 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": 1, "region": { "upper": { "closed": false, "value": "0" } } }
• { "claim": "interlacing", "from_index": 2, "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 d5ba415b6a6b3d820e30d7a55a181bfd7a0bbef2e52e742c0caaaf2bb0da456d, 3,321 bytes (1,150 compressed); download JSON

Cited by

• n-2ee8dc15b8ca5b2d (real_rooted)
• n-2ee8dc15b8ca5b2d (one_signed_coefficients)
• n-2ee8dc15b8ca5b2d (hadamard_kernel)
• n-2ee8dc15b8ca5b2d (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