Proof step e-debf4dfb42326c05

basis_map A007318 → concludes about basis_map_A007318 of Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n. 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) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n. (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 1c2c42d18cc3a69f71c59adab03989b85f6cf86cf8bd3bc072133472fdf0194f, 6,875 bytes (1,393 compressed); download JSON

Cited by

• n-4d3a36b3f07ad6a5 (real_rooted)
• n-4d3a36b3f07ad6a5 (one_signed_coefficients)
• n-4d3a36b3f07ad6a5 (hadamard_kernel)