Proof step e-5d53734a383a65e0

basis_map A007318 → concludes about basis_map_A007318 of Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2). 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, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2). (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 4e996b44182a3265ffecb09bb33a853c93debf5d8b231a98382918ef66f87553, 13,197 bytes (2,505 compressed); download JSON

Cited by

• n-e93f0e54dbc33ad5 (real_rooted)
• n-e93f0e54dbc33ad5 (one_signed_coefficients)
• n-e93f0e54dbc33ad5 (hadamard_kernel)
• n-e93f0e54dbc33ad5 (interlacing)