Proof step e-a32917d44ac940a7

basis_map A007318 → concludes about Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1). 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, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1). (sequence)
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 1]

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 2345cba956c4f310aed18ef1808c39555fb624d80a8d64479bb23397f00f2308, 14,566 bytes (2,721 compressed); download JSON

Cited by

• n-984320118b653f6b (real_rooted)
• n-984320118b653f6b (one_signed_coefficients)
• n-984320118b653f6b (hadamard_kernel)
• n-984320118b653f6b (interlacing)