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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"e6b146699de615058aa1940dfe3e6a987697bc2acebbf61eb02efec55a6393ba","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-f48bed27dbc29c5f","role":"sequence","name":"The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.","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:    9,    24,     16;\n  1:   18,    84,    128,      64;\n  2:   36,   240,    592,     640,      256;\n  3:   72,   624,   2144,    3648,     3072,     1024;\n  4:  144,  1536,   6784,   15872,    20736,    14336,     4096;\n  5:  288,  3648,  19712,   58880,   104960,   111616,    65536,     16384;\n  6:  576,  8448,  54016,  196608,   445440,   643072,   577536,    294912,     65536;\n  7: 1152, 19200, 141824,  609280,  1677312,  3067904,  3727360,   2899968,   1310720,   262144;\n  8: 2304, 43008, 360448, 1785856,  5791744, 12845056, 19726336,  20709376,  14221312,  5767168,  1048576;\n  9: 4608, 95232, 892928, 5013504, 18726912, 48857088, 90832896, 120324096, 111280128, 68419584, 25165824, 4194304;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+4*x)*P(n-1,x) for n >= 1, with P(0,x) = 9 + 24*x + 16*x^2.","mathematica":"P[0, x_] := 9 + 24*x + 16*x^2;\nP[n_, x_] := P[n, x] = Expand[(2 + 4*x)*P[n - 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