{"id":"e-c5d35f047ff7eb16","operation":{"index_shift":0,"operation":"gamma_polynomial","scale":"1"},"operation_detail":"scale 1, shift 0","conclusion_id":"n-0942305fd4723770","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":"364bfcd3dfd35d86a8cdeb049140d32eb09ca57aa9b77a470a0999aadb700f1a","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"gamma_polynomial","certificate_raw_size":6631,"certificate_compressed_size":1887,"conclusion_name":"gamma_polynomial of eulerian","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-ff270bcb802a85af","role":"sequence","name":"eulerian","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-0942305fd4723770","claim":"real_rooted"},{"node_id":"n-0942305fd4723770","claim":"one_signed_coefficients"},{"node_id":"n-0942305fd4723770","claim":"hadamard_kernel"}]}