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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"ecea8993468ad5cee2e486772573b94cfa3a564add232d58e45e1aa307809b1a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-b0d6ec97a8d172b8","role":"sequence","name":"Triangle read by rows: coefficients of the partial fraction decomposition of [d^n/dx^n] (x/(1-x))^n/n!.","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]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:      1;\n   2:      1,        4;\n   3:      5,       32,        48;\n   4:     25,      228,       672,        640;\n   5:    129,     1536,      6720,      12800,       8960;\n   6:    681,    10020,     58080,     165760,     232960,      129024;\n   7:   3653,    64032,    462000,    1756160,    3709440,     4128768,     1892352;\n   8:  19825,   403396,   3482304,   16531200,   46609920,    78059520,    71909376,    28114944;\n   9: 108545,  2514944,  25272576,  143861760,  507386880,  1135411200,  1574436864,  1237057536,   421724160;\n  10: 598417, 15554628, 178332480, 1183607040, 5011776000, 14040778752, 26027409408, 30785863680, 21086208000, 6372720640;\n","recurrence_sentence":"The row polynomials satisfy -(n-1)*P(n,x) = -(n-4)*P(n-3,x) + (7*n-23+(16*n-56)*x)*P(n-2,x) - 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