Proof step e-58e03471ce1f2a92

basis_map A013610 → concludes about basis_map_A013610 of 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. … JSON

Theorem

Geometric basis map (elementary) polytool ?

Operation

Image under a basis map u^k -> P_k

{
  "map": "A013610",
  "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 5f352863563a130d709587a7498c79657970ed8bea0de7ab1228ad8a19830ce9, 7,409 bytes (1,614 compressed); download JSON

Cited by

• n-c89ee0d57229b178 (real_rooted)
• n-c89ee0d57229b178 (one_signed_coefficients)
• n-c89ee0d57229b178 (hadamard_kernel)