{"id":"e-184795b7aaceffec","operation":{"index_shift":0,"operation":"gamma_polynomial","scale":"1"},"operation_detail":"scale 1, shift 0","conclusion_id":"n-7d3228ae4286622e","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":"a5e0726d2f236d1f8b1e27dec52dd9c731b1cf983b638eafaf4789e4ab30e0ea","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"gamma_polynomial","certificate_raw_size":6783,"certificate_compressed_size":1977,"conclusion_name":"gamma_polynomial of Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 3*n + 3.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-dfedc6a36c97eba4","role":"sequence","name":"Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 3*n + 3.","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-7d3228ae4286622e","claim":"one_signed_coefficients"},{"node_id":"n-7d3228ae4286622e","claim":"hadamard_kernel"},{"node_id":"n-7d3228ae4286622e","claim":"real_rooted"}]}