{"id":"e-ed4df29580db3bff","operation":{"index_shift":1,"operation":"gamma_polynomial","scale":"1/4"},"operation_detail":"scale 1/4, shift 1","conclusion_id":"n-3dbfcbca0c0c8d13","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":"737290ad36b92d3d209fab3273e64a3eb27748f334955e25ceb2376e5c571593","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"gamma_polynomial","certificate_raw_size":7401,"certificate_compressed_size":2120,"conclusion_name":"Triangle read by rows: T(n,k) (n >= 1, 0 <= k <= ceiling(n/2)-1) = number of permutations of [n] with k peaks.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-ff270bcb802a85af","role":"sequence","name":"eulerian","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}],"cited_by":[{"node_id":"n-3dbfcbca0c0c8d13","claim":"real_rooted"},{"node_id":"n-3dbfcbca0c0c8d13","claim":"one_signed_coefficients"},{"node_id":"n-3dbfcbca0c0c8d13","claim":"hadamard_kernel"}]}