Proof step e-5ca25d3c8ee8ed44

basis_map A007318 → concludes about basis_map_A007318 of Triangle T(n, k) read by rows: T(n, k) = Sum_{j=0..n} binomial(j, n-k) * |Stirling1(n, n-j)|. 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) read by rows: T(n, k) = Sum_{j=0..n} binomial(j, n-k) * |Stirling1(n, n-j)|. (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 773c6ca8bbff0fb7f86bc565df8190fb5c5f1f3f0194ad20ab114ea7e1b9bc65, 7,414 bytes (1,613 compressed); download JSON

Cited by

• n-67ad51294793de36 (real_rooted)
• n-67ad51294793de36 (one_signed_coefficients)
• n-67ad51294793de36 (hadamard_kernel)