{"id":"e-b66d3a1eda18e07c","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-bab8f81b674979eb","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"678ad1b34cb8791cbad761e0614ed72004b82073599ad8d7bfe48722778ee21a","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":14694,"certificate_compressed_size":2926,"conclusion_name":"shift_x_plus_one of Triangular array of the number of binary, rooted, leaf-labeled tree topologies with n leaves and k cherries, n >= 2, 1 <= k <= floor(n/2).","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-3951e6b9a140d595","role":"sequence","name":"Triangular array of the number of binary, rooted, leaf-labeled tree topologies with n leaves and k cherries, n >= 2, 1 <= k <= floor(n/2).","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}],"cited_by":[{"node_id":"n-bab8f81b674979eb","claim":"hadamard_kernel"},{"node_id":"n-bab8f81b674979eb","claim":"real_rooted"},{"node_id":"n-bab8f81b674979eb","claim":"interlacing"},{"node_id":"n-bab8f81b674979eb","claim":"one_signed_coefficients"}]}