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(elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"318d0df7635c0778fe12d7a5f20f0de8dc876603f244ceeb87ac043b585107ca","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-656046abebe1865b","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).","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]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:           1;\n  1:           4,            8;\n  2:          28,          120,           128;\n  3:         280,         1872,          4160,           3072;\n  4:        3640,        33296,        113984,         173056,          98304;\n  5:       58240,       678336,       3155584,        7328256,        8495104,         3932160;\n  6:     1106560,     15683904,      92516224,      290704896,      513163264,       482476032,       188743680;\n  7:    24344320,    407013248,    2913655552,    11576416256,    27569065984,     39351615488,     31171018752,     10569646080;\n  8:   608608000,  11733367680,   98890236672,   475884361728,  1430117289984,   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