{"id":"e-e49ea542bfed6bbf","operation":{"index_shift":1,"operation":"gamma_polynomial","scale":"1"},"operation_detail":"scale 1, shift 1","conclusion_id":"n-991b9bffcf8ddedb","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":"0baadac64d53ecdc879d2afc8184e3ce36704616bd5a3912abd5c6b727e017de","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"gamma_polynomial","certificate_raw_size":6609,"certificate_compressed_size":1890,"conclusion_name":"Triangle T(n,k) (n >= 1, 0 <= k <= floor((n-1)/2)) read by rows, where T(n,k) = (k+1)T(n-1,k) + (2n-4k)T(n-1,k-1).","claims":[{"claim":"real_rooted","from_index":0,"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-991b9bffcf8ddedb","claim":"real_rooted"},{"node_id":"n-991b9bffcf8ddedb","claim":"one_signed_coefficients"},{"node_id":"n-991b9bffcf8ddedb","claim":"hadamard_kernel"}]}