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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"030b14e6dd5c8386d36c5a5f3a19993faff6fc3265a77233524239c4d3f3c5a0","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-5a70a6e97131d437","role":"sequence","name":"Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.","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:     6,     28,      32;\n  1:    18,    132,     320,      256;\n  2:    36,    336,    1168,     1792,      1024;\n  3:   108,   1296,    6192,    14720,     17408,      8192;\n  4:   216,   3024,   17568,    54208,     93696,     86016,      32768;\n  5:   648,  10800,   76896,   303168,    714752,   1007616,     786432,     262144;\n  6:  1296,  24192,  196992,   913920,   2642176,   4874240,    5603328,    3670016,    1048576;\n  7:  3888,  82944,  784512,  4317696,  15237888,  35760128,   55803904,   55836672,   32505856,    8388608;\n  8:  7776, 181440, 1900800, 11773440,  47746560, 132471808,  254648320,  334888960,  288358400,  146800640,   33554432;\n  9: 23328, 606528, 7153920, 50526720, 237427200, 779387904, 1823719424, 3041853440, 3544186880, 2747269120, 1275068416, 268435456;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (6+28*x+32*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 6 + 28*x + 32*x^2, P(1,x) = 18 + 132*x + 320*x^2 + 256*x^3.","mathematica":"P[0, x_] := 6 + 28*x + 32*x^2;\nP[1, x_] := 18 + 132*x + 320*x^2 + 256*x^3;\nP[n_, x_] := P[n, x] = Expand[(6 + 28*x + 32*x^2)*P[n - 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