{"id":"e-a135b2bb55d9a2f7","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-31db8029f5c1d671","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":"1f089e6fad99ad23eaeb6e2e26091a0a83e1d25efdf9ce5c8080d314734677b6","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":14127,"certificate_compressed_size":2659,"conclusion_name":"shift_x_plus_one of reversal of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k peaks (1 <= k <= n).","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-6f30da2298fd30ba","role":"sequence","name":"basis_map_A007318 of narayana","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-31db8029f5c1d671","claim":"one_signed_coefficients"},{"node_id":"n-31db8029f5c1d671","claim":"hadamard_kernel"},{"node_id":"n-31db8029f5c1d671","claim":"interlacing"},{"node_id":"n-31db8029f5c1d671","claim":"real_rooted"}]}