{"id":"e-af71d95d59cd25b7","operation":{"index_shift":0,"operation":"gamma_polynomial","scale":"1"},"operation_detail":"scale 1, shift 0","conclusion_id":"n-5d8bbcaae52422ed","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":"9a35c4f0ba471dbc2066fecc55c49595afe78f7c7d7b14608ff61924b773276e","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"gamma_polynomial","certificate_raw_size":10271,"certificate_compressed_size":2374,"conclusion_name":"gamma_polynomial of Triangle read by rows: Eulerian numbers of type B, T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (2*n - 2*k + 1)*T(n-1, k-1) + (2*k - 1)*T(n-1, k).","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-1be5eefbe5bc79ec","role":"sequence","name":"Triangle read by rows: Eulerian numbers of type B, T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (2*n - 2*k + 1)*T(n-1, k-1) + (2*k - 1)*T(n-1, k).","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-5d8bbcaae52422ed","claim":"real_rooted"},{"node_id":"n-5d8bbcaae52422ed","claim":"one_signed_coefficients"},{"node_id":"n-5d8bbcaae52422ed","claim":"hadamard_kernel"}]}