{"id":"e-037c1a08736a9fd9","operation":{"index_shift":0,"operation":"gamma_polynomial","scale":"1"},"operation_detail":"scale 1, shift 0","conclusion_id":"n-97b51734d39ede6c","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":"ce49da729642bb1c9d992b199d1c5e4531c3a8efd3c11dbf180327cea22f86e7","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"gamma_polynomial","certificate_raw_size":10802,"certificate_compressed_size":2516,"conclusion_name":"Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the complete graph K(n).","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-07e5356ad37f61ea","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete graph K(n) and the complete graph K(1); in other words, K'(n) is the graph constructed from K(n) by adding for each vertex v a new vertex v' and the edge vv'.","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-97b51734d39ede6c","claim":"real_rooted"},{"node_id":"n-97b51734d39ede6c","claim":"one_signed_coefficients"},{"node_id":"n-97b51734d39ede6c","claim":"hadamard_kernel"}]}