{"id":"e-c54a32cd209a6a17","operation":{"index_shift":0,"operation":"gamma_polynomial","scale":"1"},"operation_detail":"scale 1, shift 0","conclusion_id":"n-5248ff3010d9702d","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":"8c740fdaa9c63d048e79de8b7e1378934202ce83b8e0b87ee04654bdfb3b95fb","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"gamma_polynomial","certificate_raw_size":10526,"certificate_compressed_size":2237,"conclusion_name":"reversal of Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-7f1b09730105c8f0","role":"sequence","name":"Triangle read by rows: T(n,k) = ((n+1)(n+2)/2) * binomial(n,k) (0 <= k <= n).","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-5248ff3010d9702d","claim":"real_rooted"},{"node_id":"n-5248ff3010d9702d","claim":"one_signed_coefficients"},{"node_id":"n-5248ff3010d9702d","claim":"hadamard_kernel"}]}