Proof step
Affine, reversal and Möbius substitutions (elementary)
Image under a real-rootedness preserver affine, shift 0
- Concludes
- Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
- Premises
- Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.
sequence: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
- Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.
- Theorem
- polytool ?
- Lean
- RealRooted
Polynomial.Splits.comp_of_degree_le_one - Checked
- polytool replayed · Lean not attempted
Technical details
- Step id
e-5d0c2b2d624d6b20· JSON- Operation
- Image under a real-rootedness preserver
{ "index_shift": 0, "operation": "preserver_image", "operator": "{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}" } - Claims
{ "claim": "real_rooted", "from_index": 1, "region": { "upper": { "closed": true, "value": "-1" } } }- Certificate
preserver_image, SHA-2562d83e9a2dc12b7655fcaff0010213748a7c7b8d9177e3e466237c54479a54f43, 10,138 bytes (2,647 compressed); downloadpolytool replayed the certificate when the step was written.- Cited by
n-c72653d9ab0ee8be(one_signed_coefficients)n-c72653d9ab0ee8be(hadamard_kernel)n-c72653d9ab0ee8be(real_rooted)