Proof step e-2967c46d8636e83f

basis_map A007318 → concludes about basis_map_A007318 of Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)*(2*n+3)/6)*T(n-2, k-1), read by rows. JSON

Theorem

Geometric basis map (elementary) polytool ?

Operation

Image under a basis map u^k -> P_k

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

Premises

• Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)*(2*n+3)/6)*T(n-2, k-1), 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 7f25b6825a0feb261157287575d94d2199cdb59bb01af953a63566c91e7ad6b3, 14,258 bytes (2,727 compressed); download JSON

Cited by

• n-34f0da852f498857 (real_rooted)
• n-34f0da852f498857 (one_signed_coefficients)
• n-34f0da852f498857 (hadamard_kernel)
• n-34f0da852f498857 (interlacing)