Proof step e-dc1b47ab878d31d1

preserver_image affine, shift 0 → concludes about shift_x_plus_one of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,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

• Twice Pascal's triangle A007318: T(n,k) = 2*C(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": "-1" } } }
• { "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: preserver_image, SHA-256 8a08fd57e94e47fc6da9b4dd2d7ef7662cdf1551e269c1a1b938001b471c7e86, 9,155 bytes (2,632 compressed); download JSON

Cited by

• n-416d26cbd480eee5 (real_rooted)
• n-416d26cbd480eee5 (one_signed_coefficients)
• n-416d26cbd480eee5 (hadamard_kernel)
• n-416d26cbd480eee5 (interlacing)