Proof step e-ced3e67c3d44f436

basis_map A007318 → concludes about basis_map_A007318 of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!). 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) = RisingFactorial(n + 1, n) / (k! * (n - k)!). (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 c58aec7139bcc13534c5a42c71fa0ea7677eaca3cde5a2bc22bd076109c8620c, 13,215 bytes (2,508 compressed); download JSON

Cited by

• n-0b43db26e266f660 (real_rooted)
• n-0b43db26e266f660 (one_signed_coefficients)
• n-0b43db26e266f660 (hadamard_kernel)
• n-0b43db26e266f660 (interlacing)