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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"85cdec31f1a22e456437d4c0bcb8d438e39dd8f199a471d4824391e703a72af7","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-7196be931c2ed996","role":"sequence","name":"Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,...] DELTA [3,-2,-1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.","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,      6;\n  2:     8,     20,     12;\n  3:    32,    128,    168,      72;\n  4:    64,    320,    592,     480,     144;\n  5:   256,   1664,   4288,    5472,    3456,     864;\n  6:   512,   3840,  11904,   19520,   17856,    8640,    1728;\n  7:  2048,  18432,  70656,  149504,  188544,  141696,   58752,   10368;\n  8:  4096,  40960, 178176,  440320,  676096,  660480,  400896,  138240,  20736;\n  9: 16384, 188416, 958464, 2830336, 5346304, 6698496, 5566464, 2958336, 912384, 124416;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (8+20*x+12*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 6*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 6*x;\nP[n_, x_] := P[n, x] = Expand[(8 + 20*x + 12*x^2)*P[n - 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