{"id":"e-41c16cba2e1a2f2b","operation":{"index_shift":0,"operation":"gamma_polynomial","scale":"1"},"operation_detail":"scale 1, shift 0","conclusion_id":"n-67e8e5fe38a23ea9","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":"d34130e610affc495af412e976c0975fd5b73f76973ea64b9b8f13e49d1145e3","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"gamma_polynomial","certificate_raw_size":5361,"certificate_compressed_size":1560,"conclusion_name":"reversal of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-74c1e94669ff21ff","role":"sequence","name":"basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","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-67e8e5fe38a23ea9","claim":"real_rooted"},{"node_id":"n-67e8e5fe38a23ea9","claim":"one_signed_coefficients"},{"node_id":"n-67e8e5fe38a23ea9","claim":"hadamard_kernel"}]}