{"id":"e-e0cceae8b70995e3","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-67d789c805b2e2cd","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":"eb065ef4096c24507de6011a8338c1c2a439c02638bdc930b7b2ffaf8fcb50fc","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":13345,"certificate_compressed_size":2520,"conclusion_name":"basis_map_A007318 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-67d789c805b2e2cd","claim":"real_rooted"},{"node_id":"n-67d789c805b2e2cd","claim":"one_signed_coefficients"},{"node_id":"n-67d789c805b2e2cd","claim":"hadamard_kernel"},{"node_id":"n-67d789c805b2e2cd","claim":"interlacing"}]}