{"id":"e-47e5b69025eff315","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-f3fe039b87b32aba","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":"4a959fd9c2ec51effadd987dbac92bc7886d01c8087bda487cbafb6d54f444c0","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":13197,"certificate_compressed_size":2513,"conclusion_name":"basis_map_A013609 of Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 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-dd4aa31f3d11d152","role":"sequence","name":"Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 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-f3fe039b87b32aba","claim":"real_rooted"},{"node_id":"n-f3fe039b87b32aba","claim":"one_signed_coefficients"},{"node_id":"n-f3fe039b87b32aba","claim":"hadamard_kernel"},{"node_id":"n-f3fe039b87b32aba","claim":"interlacing"}]}