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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: 2, 1;\n  1: 1, 3,  1;\n  2: 1, 5,  5,  1;\n  3: 0, 4, 11,  7,  1;\n  4: 0, 2, 13, 20,  9,   1;\n  5: 0, 0,  9, 31, 32,  11,   1;\n  6: 0, 0,  3, 29, 62,  47,  13,   1;\n  7: 0, 0,  0, 16, 74, 110,  65,  15,   1;\n  8: 0, 0,  0,  4, 56, 161, 179,  86,  17,  1;\n  9: 0, 0,  0,  0, 25, 155, 312, 273, 110, 19, 1;\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) = 2 + x, P(1,x) = 1 + 3*x + x^2, P(2,x) = 1 + 5*x + 5*x^2 + x^3, P(3,x) = 4*x + 11*x^2 + 7*x^3 + x^4.","mathematica":"P[0, x_] := 2 + x;\nP[1, x_] := 1 + 3*x + x^2;\nP[2, x_] := 1 + 5*x + 5*x^2 + x^3;\nP[3, x_] := 4*x + 11*x^2 + 7*x^3 + x^4;\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] = 2 + x; v[2] = 1 + 3*x + x^2; v[3] = 1 + 5*x + 5*x^2 + x^3; v[4] = 4*x + 11*x^2 + 7*x^3 + x^4; 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+1"],"edges":["e-9f5c84bbcac9b050"],"text":"A199881 \u2192 D+1"},"headline":"A199881 \u2192 D+1","chain":null}