Proof step e-a326dd7fdf5aa684

basis_map A013609 → concludes about basis_map_A007318 of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1]… JSON

Theorem

Geometric basis map (elementary) polytool ?

Operation

Image under a basis map u^k -> P_k

{
  "map": "A013609",
  "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 6e511d818d4ec69c71f60540240d87d6c44298fafadc5b20aabc1b195af4e6c0, 7,409 bytes (1,615 compressed); download JSON

Cited by

• n-7a2a38f140d4a893 (real_rooted)
• n-7a2a38f140d4a893 (one_signed_coefficients)
• n-7a2a38f140d4a893 (hadamard_kernel)