{"id":"e-abb1d38cae3c521a","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":"5c4aa1e02393a3229c60883a588a47f0d3a39599fe9b67ef54d2c2182d2543e6","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"gamma_polynomial","certificate_raw_size":5285,"certificate_compressed_size":1516,"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-c8c19eba9d68881f","role":"sequence","name":"binomial","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":"real_rooted"},{"node_id":"n-39194b37db596ae8","claim":"one_signed_coefficients"},{"node_id":"n-39194b37db596ae8","claim":"hadamard_kernel"}]}