Proof step e-3fcdea1a6c83c49b

basis_map A007318 → concludes about basis_map_A007318 of Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the first kind (n >= 2; 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) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the first kind (n >= 2; 1 <… (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" } } }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: basis_map, SHA-256 c874028821f21eea10f703fc0a5413e547da849b34ae2d3c22c39805e9e7dc0f, 7,400 bytes (1,613 compressed); download JSON

Cited by

• n-634bc9ca21d13c4b (real_rooted)
• n-634bc9ca21d13c4b (one_signed_coefficients)
• n-634bc9ca21d13c4b (hadamard_kernel)