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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"d96346603b8490d46465d904e70aa1d072dd718c16bb5104e0742b7400b3593d","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,      9;\n  2:     8,     30,      27;\n  3:    32,    192,     378,     243;\n  4:    64,    480,    1332,    1620,      729;\n  5:   256,   2496,    9648,   18468,    17496,     6561;\n  6:   512,   5760,   26784,   65880,    90396,    65610,    19683;\n  7:  2048,  27648,  158976,  504576,   954504,  1076004,   669222,   177147;\n  8:  4096,  61440,  400896, 1486080,  3422736,  5015520,  4566456,  2361960,   531441;\n  9: 16384, 282624, 2156544, 9552384, 27065664, 50866704, 63405504, 50545944, 23383404, 4782969;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (8+30*x+27*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 9*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 9*x;\nP[n_, x_] := P[n, x] = Expand[(8 + 30*x + 27*x^2)*P[n - 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