Proof step e-6e263990c3c1f7f9

basis_map A007318 → concludes about basis_map_A007318 of Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1. 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)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1. (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 eb7eb332d0f9250636a8b29157b39d11ebc85f8793cf25b2cf2145eda9cecb97, 7,400 bytes (1,612 compressed); download JSON

Cited by

• n-6e1be0f1b7cdfa83 (real_rooted)
• n-6e1be0f1b7cdfa83 (one_signed_coefficients)
• n-6e1be0f1b7cdfa83 (hadamard_kernel)