Proof step
Affine, reversal and Möbius substitutions (elementary)
Image under a real-rootedness preserver affine, shift 0
- Concludes
- shift_x_plus_one of Triangle read by rows, T(n,k) = (k+1)*(n+1)!*(n+k)!/((k+1)!^2*(n-k)!) with n >= 0 and 0 <= k <= n.
- Premises
- Triangle read by rows, T(n,k) = (k+1)*(n+1)!*(n+k)!/((k+1)!^2*(n-k)!) with n >= 0 and 0 <= k <= n.
sequence: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
- Triangle read by rows, T(n,k) = (k+1)*(n+1)!*(n+k)!/((k+1)!^2*(n-k)!) with n >= 0 and 0 <= k <= n.
- Theorem
- polytool ?
- Lean
- RealRooted
Polynomial.Splits.comp_of_degree_le_one - Checked
- polytool replayed · Lean not attempted
Technical details
- Step id
e-9417351d63ef338e· 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": 0, "region": { "upper": { "closed": true, "value": "-1" } } }- Certificate
preserver_image, SHA-256aceec3a18d45981cff89163a412b18e1709f3062a8a3323ae2c1751cb5097132, 9,743 bytes (2,735 compressed); downloadpolytool replayed the certificate when the step was written.- Cited by
n-07d0d70f782a78c7(real_rooted)n-07d0d70f782a78c7(one_signed_coefficients)n-07d0d70f782a78c7(hadamard_kernel)