{"id":"e-3c3f8de32007e148","operation":{"index_shift":0,"operation":"gamma_polynomial","scale":"1"},"operation_detail":"scale 1, shift 0","conclusion_id":"n-39194b37db596ae8","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":"c9ac2296cc14095180fe3f4d0c7e71f61a732e3577f3ebebe92197b72003bcd5","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"gamma_polynomial","certificate_raw_size":5329,"certificate_compressed_size":1546,"conclusion_name":"a(n) = 1 if n is a triangular number, otherwise 0.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-08f49539099949d5","role":"sequence","name":"Even-numbered rows of Pascal's triangle.","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-39194b37db596ae8","claim":"one_signed_coefficients"},{"node_id":"n-39194b37db596ae8","claim":"hadamard_kernel"},{"node_id":"n-39194b37db596ae8","claim":"real_rooted"}]}