Proof step e-fe032cf3f375eb62

basis_map A007318 → concludes about 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. JSON

Theorem

Geometric basis map (elementary) polytool ?

Operation

Image under a basis map u^k -> P_k

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

Premises

• basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1 <= k <= m*(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 228d828d30b99faaee6040438ae1243ff07706f4d6260d2e970c71cbc786f53c, 7,414 bytes (1,616 compressed); download JSON

Cited by

• n-260e022530e5976c (hadamard_kernel)
• n-260e022530e5976c (real_rooted)
• n-260e022530e5976c (one_signed_coefficients)