Proof step e-8562aab4e97e2e3f

basis_map A007318 → concludes about Triangle read by rows, s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0. 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) is the Sheffer triangle ((1 - 3*x)^(-1/3), (-1/3)*log(1 - 3*x)). A generalized Stirling1 triangle. (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 1da6f7ac835e04c2426e066aa35ec06ab0d84df6ca9a9549a9f473f24157ec78, 7,414 bytes (1,617 compressed); download JSON

Cited by

• n-8229dbdd6a4b5cd9 (one_signed_coefficients)
• n-8229dbdd6a4b5cd9 (hadamard_kernel)
• n-8229dbdd6a4b5cd9 (real_rooted)