Proof step e-484d449f4d0f99c5

basis_map A007318 → concludes about basis_map_A007318 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n. 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+4,4) * binomial(n,k), 0 <= k <= n. (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 b0cef0bec03ed03fa8aee02fd86463b2e327e96d30d577a4c1f68315eb00066a, 13,197 bytes (2,501 compressed); download JSON

Cited by

• n-fe7b03c8bdb96043 (real_rooted)
• n-fe7b03c8bdb96043 (one_signed_coefficients)
• n-fe7b03c8bdb96043 (hadamard_kernel)
• n-fe7b03c8bdb96043 (interlacing)