{"id":"e-7d7c781fa7a59405","operation":{"index_shift":0,"operation":"gamma_polynomial","scale":"1"},"operation_detail":"scale 1, shift 0","conclusion_id":"n-de8575ce50ceeb5a","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":"c244550b5f4a888564aa8219f4e01e416a3fdc41bcd56362e354edda581d69f1","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"gamma_polynomial","certificate_raw_size":9851,"certificate_compressed_size":2268,"conclusion_name":"gamma_polynomial of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-dd77bf712b1df996","role":"sequence","name":"Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","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-de8575ce50ceeb5a","claim":"real_rooted"},{"node_id":"n-de8575ce50ceeb5a","claim":"one_signed_coefficients"},{"node_id":"n-de8575ce50ceeb5a","claim":"hadamard_kernel"}]}