Proof step e-f551c1fe837853a5

basis_map A013609 → concludes about basis_map_A013609 of Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one version of) Fibonacci p… JSON

Theorem

Geometric basis map (elementary) polytool ?

Operation

Image under a basis map u^k -> P_k

{
  "map": "A013609",
  "operation": "basis_map"
}

Premises

• Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one version of) Fibonacci po… (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 9666efdf5c16abd9e7ea6d25e84c01d666176c6c2e4eccd73d91d790e44abb68, 10,694 bytes (2,406 compressed); download JSON

Cited by

• n-78b97ad8ac29ceb2 (real_rooted)
• n-78b97ad8ac29ceb2 (one_signed_coefficients)
• n-78b97ad8ac29ceb2 (hadamard_kernel)
• n-78b97ad8ac29ceb2 (interlacing)