Proof step e-ed30910ee779b0d9

basis_map A007318 → concludes about 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 <= k <= n-1; s(n,j) =… 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), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T… (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 7c85b1465afe8eb2a45d0a76cc7127a45bcf488d3f7b6e21090f3e728e4ffb52, 7,409 bytes (1,611 compressed); download JSON

Cited by

• n-914700b1060feeb2 (one_signed_coefficients)
• n-914700b1060feeb2 (hadamard_kernel)
• n-914700b1060feeb2 (real_rooted)