{"id":"e-16125ab4d60fcfd7","operation":{"index_shift":0,"operation":"gamma_polynomial","scale":"1/4"},"operation_detail":"scale 1/4, shift 0","conclusion_id":"n-5e3f8ad0b0a35d3e","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":"fafbfab63c2d005f9c6d4ee81b46c65ac61bc4983bb90c279a2f33985ab1de7c","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"gamma_polynomial","certificate_raw_size":10238,"certificate_compressed_size":2311,"conclusion_name":"Triangle read by rows: T(n,k) is the number of permutations of [n] with k increasing runs of length at least 2.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-d8312bb744732845","role":"sequence","name":"type_b_eulerian","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-5e3f8ad0b0a35d3e","claim":"real_rooted"},{"node_id":"n-5e3f8ad0b0a35d3e","claim":"one_signed_coefficients"},{"node_id":"n-5e3f8ad0b0a35d3e","claim":"hadamard_kernel"}]}