Proof step e-2276295daf37f5fe

basis_map A013609 → concludes about basis_map_A013609 of Triangle T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2), read by rows. JSON

Theorem

Geometric basis map (elementary) polytool ?

Operation

Image under a basis map u^k -> P_k

{
  "map": "A013609",
  "operation": "basis_map"
}

Premises

• Triangle T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2), read by rows. (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" } } }
• { "claim": "interlacing", "from_index": 1, "kind": "weak", "orientation": "unordered" }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: basis_map, SHA-256 1d86e1d66aef0ff495a95ee54c95754d2bbe112dea2b706e75b67156b57a86a8, 14,120 bytes (2,712 compressed); download JSON

Cited by

• n-3d92f6360b1286e9 (real_rooted)
• n-3d92f6360b1286e9 (one_signed_coefficients)
• n-3d92f6360b1286e9 (hadamard_kernel)
• n-3d92f6360b1286e9 (interlacing)