Proof step e-9f91faf8953a1d7a

basis_map A007318 → concludes about shift_x_plus_one of Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. JSON

Theorem

Geometric basis map (elementary) polytool ?

Operation

Image under a basis map u^k -> P_k

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

Premises

• Triangle read by rows: T(n, k) = binomial(n + k, 2*k). (sequence)
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]

Claims

• { "claim": "real_rooted", "from_index": 1, "region": { "upper": { "closed": true, "value": "0" } } }
• { "claim": "interlacing", "from_index": 2, "kind": "weak", "orientation": "unordered" }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: basis_map, SHA-256 0234f2ba06f53426c69ea06564ba96b9d2049e8973bfce034959815115148637, 13,345 bytes (2,515 compressed); download JSON

Cited by

• n-d0986a7acf60271a (one_signed_coefficients)
• n-d0986a7acf60271a (hadamard_kernel)
• n-d0986a7acf60271a (real_rooted)
• n-d0986a7acf60271a (interlacing)