Proof step e-95c725304ba970c4

basis_map A007318 → concludes about basis_map_A007318 of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-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 read by rows: T(n, k) = binomial(n+k-1, 2*k-1). (sequence)
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]

Claims

• { "claim": "real_rooted", "from_index": 1, "region": { "upper": { "closed": true, "value": "0" } } }
• { "claim": "interlacing", "from_index": 2, "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 a8b150c3ffce36e03b3a08d0d6e89e014c973b17fe5a28f6b0d5f2fd088b6f6a, 13,345 bytes (2,518 compressed); download JSON

Cited by

• n-e7f9db5cff35db04 (real_rooted)
• n-e7f9db5cff35db04 (one_signed_coefficients)
• n-e7f9db5cff35db04 (hadamard_kernel)
• n-e7f9db5cff35db04 (interlacing)