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DELTA (1,0,-1,1,0,0,0,0,0,0,0,...) 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: 1, 2;\n  2: 1, 4,  3;\n  3: 0, 4,  9,  4;\n  4: 0, 2, 12, 16,  5;\n  5: 0, 0,  9, 28, 25,   6;\n  6: 0, 0,  3, 28, 55,  36,   7;\n  7: 0, 0,  0, 16, 70,  96,  49,   8;\n  8: 0, 0,  0,  4, 55, 150, 154,  64,  9;\n  9: 0, 0,  0,  0, 25, 150, 287, 232, 81, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -x^2*P(n-4,x) - 2*x^2*P(n-3,x) + (2*x-x^2)*P(n-2,x) + 2*x*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 2*x, P(2,x) = 1 + 4*x + 3*x^2, P(3,x) = 4*x + 9*x^2 + 4*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[2, x_] := 1 + 4*x + 3*x^2;\nP[3, x_] := 4*x + 9*x^2 + 4*x^3;\nP[n_, x_] := P[n, x] = Expand[-x^2*P[n - 4, x] - 2*x^2*P[n - 3, x] + (2*x - x^2)*P[n - 2, x] + 2*x*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; v[3] = 1 + 4*x + 3*x^2; v[4] = 4*x + 9*x^2 + 4*x^3; for(n = 4, N, my(i = n + 1); v[i] = -x^2*v[i - 4] - 2*x^2*v[i - 3] + (2*x - x^2)*v[i - 2] + 2*x*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-f239c3b0ba932394","steps":["D"],"edges":["e-71723939a8985cd9"],"text":"A199881 \u2192 D"},"headline":"A199881 \u2192 D","chain":null}