Proof step e-fc45cb3590dab8d3

basis_map A007318 → concludes about basis_map_A007318 of Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + (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, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + (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" } } }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: basis_map, SHA-256 b84ac713a49eae3f2246eed5ade4842d1296bb08f976c5bd701a3276b4cf21e7, 7,455 bytes (1,622 compressed); download JSON

Cited by

• n-c5818ec04781ce60 (real_rooted)
• n-c5818ec04781ce60 (one_signed_coefficients)
• n-c5818ec04781ce60 (hadamard_kernel)