{"id":"e-313e6d9276f9a630","operation":{"index_shift":0,"operation":"gamma_polynomial","scale":"1"},"operation_detail":"scale 1, shift 0","conclusion_id":"n-6758fe7729845be8","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":"d72c29ea18c91554c274f815021f8f0dcd8fc6bf888721e810e3a4635d5c08e7","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"gamma_polynomial","certificate_raw_size":10287,"certificate_compressed_size":2330,"conclusion_name":"Triangular array binomial(2*n-k, k), k=0..n, n >= 0.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-14e332ad5b6d2ede","role":"sequence","name":"Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle.","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-6758fe7729845be8","claim":"one_signed_coefficients"},{"node_id":"n-6758fe7729845be8","claim":"hadamard_kernel"},{"node_id":"n-6758fe7729845be8","claim":"real_rooted"}]}