Proof step e-81fd90eeb2a8c5d0

basis_map A007318 → concludes about basis_map_A007318 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108). JSON

Theorem

Geometric basis map (elementary) polytool ?

Operation

Image under a basis map u^k -> P_k

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

Premises

• Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108). (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 fc94a501a3a1d0d7fe8a8c5d8b3f57c93d21052ad7ff7c03e824a10738d7c811, 13,217 bytes (2,512 compressed); download JSON

Cited by

• n-8d4637810467d09a (one_signed_coefficients)
• n-8d4637810467d09a (hadamard_kernel)
• n-8d4637810467d09a (interlacing)
• n-8d4637810467d09a (real_rooted)