Proof step e-b8821d58f365ef74

basis_map A007318 → concludes about Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-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) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other… (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 358a256590ca6f39ffc7ed52c34c013c40158c37ec37c05c0eda3d3cdd970cf2, 10,153 bytes (2,274 compressed); download JSON

Cited by

• n-7fd16a32c438d522 (one_signed_coefficients)
• n-7fd16a32c438d522 (hadamard_kernel)
• n-7fd16a32c438d522 (real_rooted)
• n-7fd16a32c438d522 (interlacing)