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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"a1bb29bf51176dfe8b2631d4a58ce1630c08c85854994634d20edff8b5bb6799","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-fdb186db0412847e","role":"sequence","name":"basis_map_A013610 of Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).","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,           18;\n   3:          99,          360,           324;\n   4:        1287,         6462,         10692,          5832;\n   5:       19305,       120096,        276696,        279936,         104976;\n   6:      328185,      2389122,       6865560,       9739440,        6823440,        1889568;\n   7:     6235515,     51300648,     173449836,     308629440,      304955280,      158723712,       34012224;\n   8:   130945815,   1189552878,    4565858220,    9603315288,    11959390800,     8822392992,     3571283520,     612220032;\n   9:  3011753745,  29716740864,  126426690864,  303061699584,   447925663584,   418184073216,   240942594816,   78364164096,   11019960576;\n  10: 75293843625, 797130089010, 3695568607152, 9852222925152, 16653252182112, 18517263774912, 13550878188288, 6296070809088, 1686053968128, 198359290368;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2*n+5+18*x)*P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(2*n + 5 + 18*x)*P[n - 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