Proof step e-8c1265744c32866c

basis_map A007318 → concludes about basis_map_A007318 of Triangle: row #n has n+1 terms. T(n,m) = 4^m (2n+1)! / ( (2n-2m)! (2m+1)! ). JSON

Theorem

Geometric basis map (elementary) polytool ?

Operation

Image under a basis map u^k -> P_k

{
  "map": "A007318",
  "operation": "basis_map"
}

Premises

• Triangle: row #n has n+1 terms. T(n,m) = 4^m (2n+1)! / ( (2n-2m)! (2m+1)! ). (sequence)
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]

Claims

• { "claim": "real_rooted", "from_index": 0, "region": { "upper": { "closed": true, "value": "0" } } }
• { "claim": "interlacing", "from_index": 1, "kind": "weak", "orientation": "unordered" }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: basis_map, SHA-256 3cca2af00a0fd444a90880bc377719156d6f49c88449922870d101b38ee40b4c, 13,661 bytes (2,596 compressed); download JSON

Cited by

• n-0dae7def259375c0 (real_rooted)
• n-0dae7def259375c0 (one_signed_coefficients)
• n-0dae7def259375c0 (hadamard_kernel)
• n-0dae7def259375c0 (interlacing)