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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"b193eeb207c38837cccb75586f13b8cd05edcf94a372208d68da1b3fec93f3b2","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-0444a4f352052985","role":"sequence","name":"basis_map_A007318 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:         5,          4;\n   3:        37,         60,         24;\n   4:       305,        744,        600,        160;\n   5:      2641,       8600,      10440,       5600,       1120;\n   6:     23525,      95820,     155400,     125440,      50400,       8064;\n   7:    213445,    1043700,    2118480,    2284800,    1380960,     443520,      59136;\n   8:   1961825,   11194960,   27291600,   36845760,   29752800,   14370048,    3843840,     439296;\n   9:  18205345,  118753200,  337967280,  548116800,  554067360,  357477120,  143759616,   32947200,   3294720;\n  10: 170195525, 1249158420, 4064900400, 7697437440, 9347738400, 7549686144, 4055251200, 1396961280, 280051200, 24893440;\n","recurrence_sentence":"The row polynomials satisfy -(n-1)*P(n,x) = (n-2)*P(n-2,x) - 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