Proof step e-eae4e58fb0906170

basis_map A007318 → concludes about shift_x_plus_one of Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n). 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+1-k,k) (n >= 0, 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" } } }
• { "claim": "interlacing", "from_index": 1, "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 d8088065f03d026a522e8396e7b86c60366afeab0fbaf9eb575aa68bdf2b7c24, 10,652 bytes (2,336 compressed); download JSON

Cited by

• n-aeb67fdaad2bc42e (one_signed_coefficients)
• n-aeb67fdaad2bc42e (hadamard_kernel)
• n-aeb67fdaad2bc42e (real_rooted)
• n-aeb67fdaad2bc42e (interlacing)