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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"53d0c36f9129b05af5b6048884ccb9051609839c07c86382a2d4925314c7ac29","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-ecc015942a4103af","role":"sequence","name":"basis_map_A013610 of T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.","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:           9,           12;\n   3:         121,          324,           216;\n   4:        1809,         7272,          9720,          4320;\n   5:       28401,       152280,        305640,        272160,          90720;\n   6:      458649,      3074580,       8232840,      11007360,        7348320,        1959552;\n   7:     7544041,     60693948,     203227920,     362517120,      363333600,      193995648,       43110144;\n   8:   125700129,   1179944976,    4742360784,   10578859200,    14145516000,    11337967872,     5043886848,     960740352;\n   9:  2114588641,  22686717168,  106398275760,  284904164160,   476411997600,   509432571648,   340182146304,  129699947520,   21616657920;\n  10: 35836273449, 432556027740, 2318787511920, 7245661610880, 14544257888160, 19448947469952, 17325751322880, 9914840432640, 3307348661760, 489977579520;\n","recurrence_sentence":"The row polynomials satisfy -(n-1)*P(n,x) = (n-2)*P(n-2,x) - 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