{"id":"e-50e29512dcce7c20","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-0e9006d690eb01e9","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":"69a60c0eeb1644ba75ca3c3266cdb49e1dd32a83b054c66f865b022054573dca","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":13345,"certificate_compressed_size":2523,"conclusion_name":"basis_map_A013611 of Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-318700a428d7b97b","role":"sequence","name":"Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}],"cited_by":[{"node_id":"n-0e9006d690eb01e9","claim":"interlacing"},{"node_id":"n-0e9006d690eb01e9","claim":"real_rooted"},{"node_id":"n-0e9006d690eb01e9","claim":"one_signed_coefficients"},{"node_id":"n-0e9006d690eb01e9","claim":"hadamard_kernel"}]}