{"id":"e-e2e7a7b682b47947","operation":{"index_shift":0,"operation":"gamma_polynomial","scale":"1"},"operation_detail":"scale 1, shift 0","conclusion_id":"n-cf38558d99f4fe37","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":"40c04eed59dd1c97fae4c6570133e875e3c14ab85c5a74e89dcbe8970e7a1a4a","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"gamma_polynomial","certificate_raw_size":10220,"certificate_compressed_size":2306,"conclusion_name":"Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C(n+1,2*k+1).","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-2245b35ec7cd4857","role":"sequence","name":"One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.","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-cf38558d99f4fe37","claim":"real_rooted"},{"node_id":"n-cf38558d99f4fe37","claim":"one_signed_coefficients"},{"node_id":"n-cf38558d99f4fe37","claim":"hadamard_kernel"}]}