Proof step e-0b3ba9047fae59ed

basis_map A007318 → concludes about 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(n,k) is also the un… JSON

Theorem

Geometric basis map (elementary) polytool ?

Operation

Image under a basis map u^k -> P_k

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

Premises

• Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 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" } } }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: basis_map, SHA-256 3addd622b3f139c2f54a644ebbe598fb6c42c0cf1baa8e47f8094996b0408281, 7,412 bytes (1,612 compressed); download JSON

Cited by

• n-efe23a76953cf69d (one_signed_coefficients)
• n-efe23a76953cf69d (hadamard_kernel)
• n-efe23a76953cf69d (real_rooted)