Proof step e-4d7e15a01b4f3b87

preserver_image affine, shift 0 → concludes about basis_map_A007318 of Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k. 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\":\"1\"}"
}

Premises

• Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k. (sequence)
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4, 0)

Claims

• { "claim": "real_rooted", "from_index": 1, "region": { "lower": { "closed": false, "value": "-5" }, "upper": { "closed": false, "value": "-1" } } }
• { "claim": "interlacing", "from_index": 2, "kind": "weak", "orientation": "unordered" }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: preserver_image, SHA-256 3d37b2ffae35b089b3b25f855ded8e59b8ad7ba6de6896e71f4eb82d1102cdcf, 18,051 bytes (3,499 compressed); download JSON

Cited by

• n-67d789c805b2e2cd (one_signed_coefficients)
• n-67d789c805b2e2cd (hadamard_kernel)
• n-67d789c805b2e2cd (interlacing)
• n-67d789c805b2e2cd (real_rooted)