Proof step e-61d6f3dd155b69fd

preserver_image affine, shift 0 → concludes about a(n) = 1 if n is of the form m(m+3)/2, otherwise 0. JSON

Theorem

Affine, reversal and Möbius substitutions (elementary) polytool ?

Lean (RealRooted): Polynomial.Splits.comp_of_degree_le_one

Operation

Image under a real-rootedness preserver

{
  "index_shift": 0,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"affine\",\"slope\":\"-1\",\"intercept\":\"0\"}"
}

Premises

• basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0 (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": { "lower": { "closed": true, "value": "0" } } }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: preserver_image, SHA-256 583d6945f63ec19e9c67ab28cf8869084632f879936234a5de327f6185abcc00, 4,778 bytes (1,488 compressed); download JSON

Cited by

• n-6fea5aff4f44fa54 (real_rooted)
• n-6fea5aff4f44fa54 (one_signed_coefficients)