{"id":"e-250eeed7ad07d148","operation":{"index_shift":0,"operation":"gamma_polynomial","scale":"1"},"operation_detail":"scale 1, shift 0","conclusion_id":"n-e3d901814302c059","premise_count":1,"theorem":"gamma criterion (RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff): if P = sum_j gamma_j x^j (1 + x)^(d - 2j) with deg Gamma <= d/2, then P is nonzero and real-rooted with roots in (-inf, 0] if and only if Gamma(z) = sum_j gamma_j z^j is; Gamma(c z) with c != 0 has the roots of Gamma divided by c","lean_route":null,"lean_declarations":[],"certificate_hash":"c510169b4a05033d80a373255159ffad183c5fbaac396c59b2a69d812e060102","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"gamma_polynomial","certificate_raw_size":10504,"certificate_compressed_size":2405,"conclusion_name":"Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pieces of 1 X 1 tiles (0 <= k <= floor(n/2)).","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-f25cd955a876cde3","role":"sequence","name":"Pascal-(1,2,1) array read by antidiagonals.","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-e3d901814302c059","claim":"real_rooted"},{"node_id":"n-e3d901814302c059","claim":"one_signed_coefficients"},{"node_id":"n-e3d901814302c059","claim":"hadamard_kernel"}]}