Proof step e-74760f8a95c660e1

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

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: preserver_image, SHA-256 e9a44ecf57d62a5c24bf39c7aa88bb4ad9d0f8d3ec30166c7756c12dc13fbd60, 7,256 bytes (2,001 compressed); download JSON

Cited by

• n-ff3909e492eb580d (real_rooted)
• n-ff3909e492eb580d (one_signed_coefficients)
• n-ff3909e492eb580d (hadamard_kernel)